O-MINIMAL HAUPTVERMUTUNG FOR POLYHEDRA II
MASAHIRO SHIOTA
Hilbert initiated the finitary standpoint in foundations of mathematics. From this standpoint we consider manifolds and introduce a ``tame'' manifold, whose term is derived from \cite{v3}. A tame manifold is such that any subset of a line obtained from it by elementary operations has only a finite number of connected components. In this article and a forthcoming one entitled o-minimal Hauptvermutung for polyhedra III we will show that a tame $C^0$ manifold is tamely homeomorphic to the interior of a compact PL (=piecewise linear) manifolds possibly with boundary and such a PL manifold possibly with boundary is unique up to PL homeomorphisms in the sense that if $M_1$ and $M_2$ are such PL manifolds possibly with boundary then $M_1$ and $M_2$ are PL homeomorphic.
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\begin{document}
\title{O-MINIMAL HAUPTVERMUTUNG FOR POLYHEDRA II}
\author{MASAHIRO SHIOTA}
 \maketitle
\begin{abstract}
Hilbert initiated the finitary standpoint in foundations of mathematics.
From this standpoint we consider manifolds and introduce a ``tame'' manifold, whose term is derived from \cite{v3}.
A tame manifold is such that any subset of a line obtained from it by elementary operations has only a finite number of connected components.
In this article and a forthcoming one entitled o-minimal Hauptvermutung for polyhedra III we will show that a tame $C^0$ manifold is tamely homeomorphic to the interior of a compact PL (=piecewise linear) manifolds possibly with boundary and such a PL manifold possibly with boundary is unique up to PL homeomorphisms in the sense that if $M_1$ and $M_2$ are such PL manifolds possibly with boundary then $M_1$ and $M_2$ are PL homeomorphic.
\end{abstract}
\footnotetext{2010 {\it Mathematics Subject Classification}, 03C64, 57Q05, 57Q25.\\
\hspace{5.5mm}{\it Key words and phrases}. PL manifolds; compactification; Hauptvermutung; o-minimal structures.}
\section{Introduction}
Let $\N$ denote non-negative integers.
For a subset $X$ of a Euclidean space $\R^m$, let $S_n(X),\ n\in\N$, denote the smallest families of subsets of $\R^n$ such that $X\subset S_m(X)$,\\
(i) {\it $S_n(X)$ is a boolean algebra of subsets of $\R^n$,}\par\noindent
(ii) {\it if $Y\in S_n(X)$, then $\R\times Y$ and $Y\times\R$ are elements of $S_{n+1}(X)$,}\par\noindent
(iii) {\it every linear subspace of $\R^n$ is an element of $S_n(X)$,}\par\noindent 
(iv) {\it the set $\{(x,y)\in\R^2:x<y\}$ is an element of $S_2(X)$,} and\par\noindent
(v) {\it if $Y\in S_{n+1}(X)$, then the image of $Y$ under the projection of $\R^{n+1}$ onto the first $n$ coordinates is an element of $S_n(X)$}.\vskip1mm\noindent
We call $X$ {\it tame} if\\
(vi) {\it any element of $S_1(X)$ has only a finite number of connected components,}\vskip1mm\noindent
and we call a map $f:X\to Y$ between subsets of Euclidean spaces {\it tame} if $\graph f$ is tame.
A {\it tame} $C^0$ manifold is both a tame set and a $C^0$ manifold having an atlas $\{(U_\alpha,\psi_\alpha)\}_{\alpha\in A}$ such that $\psi_\alpha:U_\alpha\to\R^n$ are tame.
Examples of a tame set and a tame $C^0$ manifold are an algebraic set and a non-singular algebraic variety over $\R$, and an example of a non-tame set is the graph of the function $y=\sin x$.\par
 In this article and o-minimal Hauptvermutung for polyhedra III we will prove the following.\vskip2mm\noindent
{\bf Theorem 1.} {\it  A compact tame topological manifold $M$ admits a unique PL manifold structure, i.e.\ $M$ is tamely homeomorphic to a compact PL manifold $M_1$ and $M_1$ is unique up to PL homeomorphisms in the sense that if $M_2$ is another PL manifold tamely homeomorphic to $M$ then $M_1$ and $M_2$ are PL homeomorphic.
A non-compact tame topological manifold is tamely homeomorphic to the interior of a compact PL manifold with boundary $M_1$, and $M_1$ is unique up to PL homeomorphisms.}\vskip2mm
 We apply a theory of o-minimal structures---a notion of model theory---to prove this theorem.
Model theorists Knight, Pillay and Steinhorn defined o-minimal structures \cite{P-S} and \cite{K-P-S}, and o-minimal structures have been extensively studied ever since.
Let $R$ denote an ordered field.
An {\it open interval} of $R$ is a subset of $R$ of the form $\{x\in R:a<x<b\}$ for some $a,b\in R\cup\{\pm\infty\}$.
Hence when $R$ is the rational number field $\Q$, we do not call the set $\{x\in\Q:-2^{1/2}<x<2^{1/2}\}$ an interval of $\Q$ because the end points are not elements of $\Q$.
An {\it o-minimal structure} over $R$ is a sequence $\{S_n:n\in\N\}$ such that for each $n\in\N$,\\
(i) {\it $S_n$ is a boolean algebra of subsets of $R^n$,}\par\noindent
(ii) {\it if $X\in S_n$, then $R\times X$ and $X\times R$ are elements of $S_{n+1}$,}\par\noindent
(iii) {\it every linear subspace of $R^n$ is an element of $S_n$,}\par\noindent 
(iv) {\it the set $\{(x,y)\in R^2:x<y\}$ is an element of $S_2$,}\par\noindent
(v) {\it if $X\in S_{n+1}$, then the image of $X$ under the projection of $R^{n+1}$ onto the first $n$ coordinates is an element of $S_n$,} and\par\noindent
(vi) {\it an element of $S_1$ is a finite union of points and open intervals.}\vskip1mm
 The simplest example of an o-minimal structure is the families of {\it semilinear} sets in $R^n,\ n\in\N$, which are defined to be the finite unions of sets $\{x\in R^n:f_i(x)\dag_i0,\,i\in I\}$, where $\dag_i$ is either = or $>$, $I$ is a finite set and $f_i$ are linear functions on $R^n$ with coefficients in $R$.
An element of $S_n$ is called {\it definable}, and a map between definable sets is called {\it definable} if its graph is definable.
We endow a topology on $R$ and then $R^n$ as in the case of real numbers so that an open set in $R$ is a union of open intervals, and we naturally define a $C^0$ manifold, a polyhedron and a PL manifold over $R$.
However, we are interested only in definable ones.
A {\it definable} $C^0$ or $C^1$ {\it manifold} is both a definable subset of some $R^n$ and a $C^0$ or $C^1$ manifold having an atlas $\{(U_\alpha,\psi_\alpha)\}_{\alpha\in A}$ of definable class, i.e.~$U_\alpha$ and $\psi_\alpha$ are definable.
A {\it definable polyhedron} and a {\it definable PL manifold} are a polyhedron and a PL manifold included and definable in some $R^n$.
See fundamental properties of definable sets in \cite{S5} and \cite{v3}.
In this series we study topology of definable sets and definable $C^0$ maps, and one of the main theorems is the following.\vskip2mm\noindent
{\bf Theorem 2.} {\it A definable $C^0$ manifold is definably homeomorphic to the interior of a unique compact PL manifold possibly with boundary.}\vskip2mm
 Here we call a definable set in $R^n$ {\it compact} if it is bounded and closed in $R^n$, a polyhedron in $R^n$ a {\it compact polyhedron} if it is a finite union of simplexes in $R^n$ and a map between compact polyhedra {\it PL} if its graph is not only a polyhedron but also a compact polyhedron.
We have seen in \cite{S6} the facts: there exists a polyhedron closed and bounded in $R^n$ which cannot be a finite union of simplexes; a compact definable set or a compact polyhedron is not compact in the usual sense unless $R$ is the real number field $\R$; a polyhedron in $R^n$ is a compact polyhedron if it is a compact definable set.\par
 In \cite{S6} we showed that a compact definable set is definably homeomorphic to a unique compact polyhedron.
In the present paper we prove Theorems 1 and 2 by using results in \cite{S6} and strengthening the axiom (iii) to\vskip1mm\noindent
(iii)$'$ {\it every algebraic set in $R^n$ $($or $\R^n)$ is an element of $S_n$ $($or $S_n(X))$.}\vskip1mm\noindent
We postpone proving them under (iii) until the forthcoming paper.
There we need some fundamental arguments on o-minimal structures which are partially explained in Chapter V of \cite{S5}.
Note that when we admit (iii)$'$, the simplest example of an o-minimal structure is the families of {\it semialgebraic} sets in $R^n,\ n\in\N$.
See Definition 10 in Sect.~2 for the definition of a semialgebraic set.\par
 We state Theorem 2 over general $R$ but not $\R$.
One reason is that we construct a theory of manifolds and maps after starting from axioms as few as possible and we do not want to apply special properties of real numbers as axiom.
The second reason is that we prove Theorem 2 by a machinery whose input is a definable $C^0$ manifold, the operations are only the operations which appear in (i), (ii) and (v), and the output is the couple of a PL manifold possibly with boundary and a definable homeomorphism between the $C^0$ manifold and the interior of the PL manifold possibly with boundary (see \cite{S6} for the details).
We call such a proof {\it constructive}.
We cannot constructively prove many properties of real numbers.
An example is compactness of a closed bounded set in $\R^n$ in the usual sense.
To be precise, given an open covering of a closed bounded subset of $\R^n$, we have no constructive method to choose a finite subcovering.
Another example is the Archimedean property of real numbers.
Given positive real numbers $x$ and $y$, there is no constructive method to find a natural number $n$ such that $x<n y$.
Our proof in the present paper is always constructive.
The third reason is that if a statement is constructively proved over some ordered field with some o-minimal structure, then in many cases it holds over any ordered field with any o-minimal structure.
This is the case for Theorem 2.\par
 In a machinery we admit only finite repetition of operations.
This causes some special phenomena of definable manifolds and definable homeomorphisms as in Theorem 2, which never occur for general manifolds and homeomorphisms.
Let us consider four kinds of manifolds: first, a topological manifold which does not admit a PL manifold structure \cite{K-S1}; secondly, two PL manifolds which are homeomorphic but not PL homeomorphic (a counterexample to the Hauptvermutung) \cite{K-S1}; thirdly, a non-compact $C^0$ (or PL) manifold which cannot be (PL) homeomorphic to the interior of a compact $C^0$ (PL) manifold with boundary (a non-compactifiable manifold, e.g.~$\R^2-\Z^2$); fourthly, two compact $C^0$ (PL) manifolds with boundary which are not (PL) homeomorphic but whose interiors are (PL) homeomorphic (e.g.~a disc and a compact contractible manifold which is not a disc \cite{Ma}).
In o-minimal topology of definable manifolds and definable $C^0$ maps from Hilbert's standpoint there are no such wild phenomena.
This is what we would argue in the series and, in particular, in this paper.\par
 Tame sets are similar but different to definable sets in an o-minimal structure over $\R$.
A definable set in an o-minimal structure over $\R$ is clearly tame, hence a set in a Euclidean space is tame if and only if the set is definable in some o-minimal structure over $\R$, and, moreover, a map $f:X\to Y$ between subsets of Euclidean spaces is tame if and only if $X,\ \Ima f$ and $f:X\to\Ima f$ are definable in some o-minimal structure over $\R$.
However, for two tame sets $X$ and $Y$ in Euclidean spaces there does not necessarily exist an o-minimal structure over $\R$ where $X$ and $Y$ are both definable \cite{R-S-W}.
Hence it is not trivial that a tame $C^0$ manifold is a definable $C^0$ manifold in some o-minimal structure over $\R$.
We will prove this.
Then Theorem 1 follows from Theorem 2.\par
 Theorem 2 in the compact case is an easy consequence of Triangulation theorem of definable sets and Uniqueness theorem of definable triangulation.
The first theorem is well-known, and the second was proved in \cite{S6}.
We explain both the theorems in the next section.
In the non-compact case Uniqueness theorem is false.
We will choose triangulations of non-compact definable sets, which we call {\it standard} triangulations, so that uniqueness holds (Theorem 4).
For the proof of Theorem 4 we use Triangulation theorem of definable $C^0$ functions, which also was proved in \cite{S6} and was the key lemma of the proof of Uniqueness theorem of definable triangulation.
\section{Definitions and facts quoted from \cite{S6}}
In this section we make a list of definitions and facts which are given in \cite{S6} and used in this paper.\par
 We can define most terminology of PL topology over $R$ also, e.g.~a simplex, a simplicial complex and a cell complex.
We do not give definitions of those terms which do not cause confusion.\vskip1mm\noindent
{\it Definition 1.} A {\it full} subcomplex $L$ of a simplicial complex $K$ is a subcomplex such that each $\sigma\in K$ whose vertices are all in $L$ is a simplex in $L$.\vskip1mm\noindent
{\it Definition 2.}  A {\it cell} $\sigma$ in $R^n$ is a {\it bounded} semilinear subset of $R^n$ of the form $\{x\in R^n:f_i(x)\dag_i0,\,i\in I\}$, where $\dag_i$ is now either = or $\ge$, $I$ is a finite set and $f_i$ are linear functions.
The interior and boundary of $\sigma$ are denoted by $\Int\sigma$ and $\partial\sigma$, respectively.\vskip1mm\noindent
{\it Definition 3.} For a union $X$ of cells in a cell complex $L$, set $L|_X=\{\sigma\in L:\sigma\subset X\}$.\vskip1mm\noindent
{\it Definition 4.} A {\it derived} subdivision of a cell complex $L$ is a simplicial subdivision $L'$ of $L$ defined by induction such that for each $\sigma\in L$, the restriction $L'|_\sigma$ is the family of the cones with vertex one point in $\Int\sigma$ and bases $L'|_{\partial\sigma}$.\vskip1mm\noindent
{\it Definition 5.} Let $X\supset Y$ be compact polyhedra.
A {\it collar} on $Y$ in $X$ is a PL embedding $\phi:Y\times[0,\,1]\to X$ such that $\phi(\cdot,0)=\id$ and $\phi(Y\times[0,\,1))$ is an open neighborhood of $Y$ in $X$.\vskip1mm\noindent
{\it Definition 6.} A compact polyhedral neighborhood $U$ of $Y$ in $X$ is called a {\it regular} neighborhood of $Y$ in $X$ if there exist compact polyhedra $X_1\supset U_1\supset Y_1$, a PL homeomorphism $h:(X;U,Y)\to(X_1;U_1,Y_1)$ and a simplicial decomposition $K_1$ of $X_1$ such that $K_1|_{Y_1}$ is a full subcomplex of $K_1$ and $U_1=\cup_{x\in Y_1}|\st(x,K'_1)|$.
Here a PL homeomorphism $h:(X;U,Y)\to(X_1;U_1,Y_1)$ is a PL homeomorphism from $X$ to $X_1$ carrying $U$ and $Y$ to $U_1$ and $Y_1$, respectively, and $K'_1,\ \st(x,K'_1)$ and $|K_1|$ denote a derived subdivision of $K_1$, the star of $x$ in $K'_1$ and the underlying polyhedron of $K_1$, respectively.\vskip1mm\noindent
{\it Definition 7.} Let $x*X$ denote the {\it cone} with vertex $x$ and base $X$.\vskip1mm\noindent
(8) {\it of} \cite{S6}, Sect.~2. {\it A PL map $g:X\to Y$ is extended to a PL map $g^*:x*X\to y*Y$ so that $g^*(x)=y$ and $g^*$ is linear on the segment with ends $x$ and any point of $X$, and $g^*$ is a PL homeomorphism if $g$ is so} (the Alexander trick).\vskip1mm\noindent
(9.1) {\it of} \cite{S6}, Sect.~2. {\it Given a finite simplicial complex $K$, a full subcomplex $L$ and two derived subdivisions $K_1$ and $K_2$ of $K$ such that $K_1|_{|L|}=K_2|_{|L|}$, there is an isomorphism $\tau:K_1\to K_2$ such that $\tau=\id$ on $|L|\cup|K^0|$ and }
$$
\tau(\cup_{x\in|L|}|\st(x,K_1)|)=\cup_{x\in|L|}|\st(x,K_2)|.
$$\vskip1mm\noindent
(9.2) {\it of} \cite{S6}, Sect.~2. {\it Given $K$ and $L$ as above, a full subcomplex $M$ of $L$ and a subdivision $K_1$ of $K$, there are derived sub\-divisions $K'$ of $K$ and $K'_1$ of $K_1$ such that }
$$
\cup_{x\in|L|-|M|}|\st(x,K')|=\cup_{x\in|L|-|M|}|\st(x,K'_1)|.
$$
\vskip1mm\noindent
(9.4) {\it of} \cite{S6}, Sect.~2. {\it For a compact PL manifold with boundary $X$, a regular neighborhood of $\partial X$ in $X$ is a collar on $\partial X$ in $X$}.\vskip1mm\noindent
(13) {\it of} \cite{S6}, Sect.~2. {\it Let $\phi$ and $\psi$ be non-negative PL functions on a compact polyhedron $X$ such that $\phi^{-1}(0)=\psi^{-1}(0)$, and let $K$ be a simplicial decomposition of $X$.
Then there exists a PL isotopy $\tau_t,\ 0\le t\le1$, of $X$ preserving $K$ such that $\tau_t=\id$ on the zero set of $\phi$ and $\phi\circ\tau_1=\psi$ on some neighborhood of the zero set of $\phi$ in $X$}.\vskip1mm\noindent
{\it Definition 8.} An {\it open simplex (cell)} is the interior of a simplex (cell), which is never used in PL topology.\vskip1mm\noindent
{\it Definition 9.} A {\it semilinear} map is a map between semilinear sets whose graph is semilinear.\vskip1mm\noindent
{\it Definition 10.} A {\it semialgebraic} set in $R^n$ is the finite union of sets $\{x\in R^n:f_i(x)\dag_i0,\,i\in I\}$, where $\dag_i$ is either = or $>$, $I$ is a finite set and $f_i$ are polynomial functions on $R^n$ with coefficients in $R$.\vskip1mm\noindent
{\it Definition 11.} A {\it semialgebraic} map between semialgebraic sets is naturally defined.\vskip1mm\noindent
{\it Definition 12.} Let $X$ and $Y$ be compact definable sets in $R^n$.
Two definable $C^0$ maps $f,\,g:X\to Y$ are {\it definably isotopic} or $f$ is {\it definably isotopic} to $g$ if there exists a definable $C^0$ map $F:X\times[0,\,1]\to Y$ such that $F(\cdot,0)=f(\cdot)$, $F(\cdot,1)=g(\cdot)$ and $F(\cdot,t)$ is a PL embedding for each $t\in[0,\,1]$.
We write $F(\cdot,t)$ as $f_t(\cdot)$, $0\le t\le 1,$ and call it a {\it definable homotopy} of $f_0$ to $f_1$.\vskip1mm\noindent
{\it Definition 13.} A {\it definable isotopy} of $X$ (to $f$) is a definable $C^0$ map $F:X\times[0,\,1]\to X$ such that $F(\cdot,0)=\id$ and $F(\cdot,t)$ is a homeomorphism of $X$ for each $t\in[0,\,1]$ ($F(\cdot,1)=f(\cdot)$).
Note that a definable isotopy $F:X\times[0,\,1]\to X$ of the identity map does not mean a definable isotopy of $X$.
In the former case, $F(\cdot,t)$ is not necessarily a homeomorphism of $X$; it is an embedding.\vskip1mm\noindent
{\it Definition 14.} In order to distinguish them we call a definable $C^0$ map $F:X\times[0,\,1]\to Y$ a {\it definable isotopy through homeomorphisms} if for each $t$, the map $F(\cdot,t)$ is a definable homeomorphism from $X$ to $Y$, and call definable $C^0$ maps $f,g:X\to Y${\it definable isotopic through homeomorphisms} if there exists a definable isotopy of $f$ to $g$ through homeomorphisms.\vskip1mm\noindent
{\it Definition 15.} We call a point $s$ of $R^n$ with $|s|=1$ a {\it singular direction} for a definable subset $X$ of $R^n$ if the set $X\cap(a+R s)$ has interior points in the line $a+R s$ for some point $a$ in $X$, where $a+R s$ denotes the set $\{a+b s:b\in R\}$.\vskip1mm\noindent
{\it Definition 16.} The {\it Alexander trick} is the following extension of a map.
Let $X$ and $Y$ be compact definable sets in $R^n$, let $x$ and $y$ points in $R^n$ such that there exist the cones with vertices $x$ and $y$ and bases $X$ and $Y$, respectively, and let $g:X\to Y$ be a definable $C^0$ map.
Then $g$ is extended to a definable map $g^*:x*X\to y*Y$ so that $g^*(x)=y$ and $g^*$ is linear on each segment with ends $x$ and a point of $X$, and $g^*$ is a homeomorphism if $g$ is a homeomorphism.\vskip1mm\noindent
{\it Fact 17.} {\it A definable set $E$ in $R^n$ is a finite disjoint union of definable $C^1$ manifolds.}\vskip1mm\noindent
{\it Definition 18.} We call $\{E_1,...,E_k\}$ a {\it definable $C^1$ stratification} of $E$ and simply write it as $\{E_i\}_i$.\vskip1mm\noindent
{\it Definition 19.} For simplicity of notation and without loss of generality we assume a definable $C^1$ stratification $\{E_i\}_i$ satisfies the {\it frontier condition} (i.e.~if $E_i\cap(\overline{E_{i'}}-E_{i'})\not=\emptyset$ then $E_i\subset\overline{E_{i'}}$) and the condition that each $E_i$ is {\it definably connected} (i.e.~$E_i$ does not contain a proper definable open and closed subset).\vskip1mm\noindent
{\it Definition 20.} The stratification $\{E_i\}_i$ is {\it compatible} with a finite family of definable sets $\{X_j\}_j$ in $R^n$ if each $E\cap X_j$ is the union of some $E_i$.\vskip1mm\noindent
{\it Fact 21.} {\it Given a definable $C^0$ map $g:X\to Y$ between definable sets, there exist definable $C^1$ stratifications $\{X_i\}_i$ of $X$ and $\{Y_j\}_j$ of $Y$ such that for each $i$, the restriction $g|_{X_i}$ is a surjective $C^1$ submersion onto some $Y_j$.}\vskip1mm\noindent
{\it Definition 22.} Then we write $g:\{X_i\}_i\to\{Y_j\}_j$ and call it a {\it definable} $C^1$ {\it stratification} of $g$.\vskip1mm\noindent
{\it Fact 23.} {\it Moreover, given finite families of definable sets $\{A_\nu\}_\nu$ of $X$ and $\{A'_{\nu'}\}_{\nu'}$ of $Y$, we can choose a definable $C^1$ stratification $g:\{X_i\}_i\to\{Y_j\}_j$ so that $\{X_i\}_i$ and $\{Y_j\}_j$ are compatible with $\{A_\nu\}_\nu$ and $\{A'_{\nu'}\}_{\nu'}$, respectively.}\vskip1mm\noindent
{\it Definition 24.} Then we say that $g:\{X_i\}_i\to\{Y_j\}_j$ is {\it compatible} with $(\{A_\nu\}_\nu,\{A'_{\nu'}\}_{\nu'})$.\vskip2mm\noindent
 {\bf Triangulation theorem of definable sets.} {\it Given a finite simplicial complex $K$ in $R^n$ and a finite number of compact definable sets $\{X_i\}_i$ in $|K|$, there exists a definable homeomorphism $\tau$ of $|K|$ preserving $K$ such that the $\{\tau^{-1}(X_i)\}_i$ are polyhedra, and, moreover, there exists a definable isotopy $\tau_t,\ 0\le t\le 1$, of $|K|$ preserving $K$ such that $\tau_1=\tau$.}\vskip1mm\noindent
{\it Definition 25.} We call $\tau:\{\tau^{-1}(X_i)\}_i\to\{X_i\}_i$ a {\it definable triangulation} of $\{X_i\}_i$.\vskip2mm\noindent
{\bf Triangulation theorem of definable $C^0$ functions.}\\
{\it $(1)$ Let $f$ be a definable continuous $\R$-valued function defined on a compact polyhedron $X$ in $R^n$ and let $P$ be a cellular decomposition of $X$.
Then there exists a definable homeomorphism $\pi$ of $X$ preserving $P$ such that $f\circ\pi$ is PL.\\
$(2)$ Such a $\pi$ is the finishing homeomorphism of some definable isotopy of $X$ preserving $P$.\\
$(3)$ Moreover, $\pi$ is unique in the following sense.
Let $\pi'$ be a definable homeomorphism of $X$ preserving $P$ such that $f\circ\pi'$ is PL.
Then there exists a definable isotopy $\omega_t,\ 0\le t\le 1$, of $\pi^{-1}\circ\pi'$ through homeomorphisms preserving $P$ such that $\omega_1$ is PL and $f\circ\pi\circ\omega_t=f\circ\pi'$ for $t\in[0,\,1]$.}\vskip2mm\noindent
{\bf Uniqueness theorem of definable triangulation.} {\it If two families of compact polyhedra $(X_1;X_{1,1},\allowbreak...,X_{1,k})$ and $(X_2;X_{2,1},...,X_{2,k})$ are definably homeomorphic then they are PL homeomorphic.}\vskip2mm
 Here the notation $(X_1;X_{1,1},...,X_{1,k})$ indicates that $X_{1,i}$ are subsets of $X_1$, we say that $(X_1;X_{1,1},...,\allowbreak X_{1,k})$ and $(X_2;X_{2,1},...,X_{2,k})$ are definably (or PL) homeomorphic if there is a definable (or PL) homeomorphism from $X_1$ to $X_2$ carrying each $X_{1,i}$ to $X_{2,i}$, and we write $(X_1;X_{1,1},...,X_{1,k})$ as $(X_1;X_{1,i})$.\vskip1mm
Let us follow the proof of Triangulation theorem of definable $C^0$ functions (1) of \cite{S6}.
Let $p:R^n\times R\to R$ and $q_1:R^n\times R\to R^n$ be the projections and let $R^n\times R\stackrel{p_1}{\longrightarrow}R^{n-1}\times R\longrightarrow\cdots\stackrel{p_n}{\longrightarrow}R$ be the projections ignoring the respective first factors.
Set $A=\graph f$ and $A_t=\{x\in R^n:(x,t)\in A\}$ for each $t\in R$.
For simplicity of notation, we assume that $A$ is a compact definable subset of $R^n\times R$ of local dimension $n$ or $n+1$.
There exists a definable $C^1$ stratification $\{A_j\}_j$ of $A$ compatible with $\{\sigma\times R:\sigma\in P\}$ such that the $\{p_1(A_j)\}_j,...,\{p_n\circ\cdots\circ p_1(A_j)\}_j$ are definable $C^1$ stratifications of $p_1(A),...,p_n\circ\cdots\circ p_1(A)$, respectively, and the $p_1|_A:\{A_j\}_j\to\{p_1(A_j)\}_j,...,\,p_n|_{p_{n-1}\circ\cdots\circ p_1(A)}:\{p_{n-1}\circ\cdots\circ p_1(A_j)\}_j\to\{p_n\circ\cdots\circ p_1(A_j)\}_j$ are definable $C^1$ stratifications of $p_1|_A:A\to p_1(A),...,\,p_n|_{p_{n-1}\circ\cdots\circ p_1(A)}:p_{n-1}\circ\cdots\circ p_1(A)\to p_n\circ\cdots\circ p_1(A)$, respectively.
We simply write $\{A_j\}_j\stackrel{p_1}{\longrightarrow}\{p_1(A_j)\}_j\stackrel{p_2}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}\{p_n\circ\cdots\circ p_1(A_j)\}_j$ and $A\stackrel{p_1}{\longrightarrow}p_1(A)\stackrel{p_2}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}p_n\circ\cdots\circ p_1(A)$.) 
Then, since $\dim p^{-1}_k(a)=1\ (k=1,...,n)$ for each $a\in p_k\circ\cdots\circ p_1(A)$, each $p_{k-1}\circ\cdots\circ p_1(A_j)$ is described by definable $C^1$ functions on $p_k\circ\cdots\circ p_1(A_j)$ as follows.
There exist definable $C^1$ functions $\phi_{k,j}$ and $\psi_{k,j}$ on $p_k\circ\cdots\circ p_1(A_j)$ such that 
$$
p_{k-1}\circ\cdots\circ p_1(A_j)=\left\{
\begin{array}{l}\!
\{(x_k,x',t)\in R\times(p_k\circ\cdots\circ p_1(A_j)):\phi_{k,j}(x',t)<x_k<\psi_{k,j}(x',t)\}\\\!\text{or}\ \{(x_k,x',t)\in R\times(p_k\circ\cdots\circ p_1(A_j)):x_k=\phi_{k,j}(x',t)\},\end{array}
\right.
$$
where $x'\in R^{n-k}$.
Then we need the following two conditions.\vskip1mm\noindent
{\it Condition} (**) {\it Each of $\phi_{k,j}$ and $\psi_{k,j}$ is extended to a definable $C^0$ function on $p_k\circ\cdots\circ p_1(\overline{A_j})$.}\vskip2mm
Let $R^n\stackrel{r_1}{\longrightarrow}\cdots\stackrel{r_{n-1}}{\longrightarrow}R$ denote the projections ignoring the respective first factors, and let $\{A_{k,j}\}_j$ and $P_k$ denote $\{p_{k-1}\circ\cdots\circ p_1(A_j)\}_j$ and $\{r_{k-1}\circ\cdots\circ r_1(\sigma):\sigma\in P\}$, respectively.\vskip2mm\noindent
{\it Condition} (***) {\it Given $\sigma\in P_k$ and $A_{k,j}$ such that $A_{k,j}\subset\Int\sigma\times R$ and $\dim A_{k,j}\le\dim\sigma$, the restrictions $q_k|_{A_{k,j}}$ and $q_k|_{\overline{A_{k,j}}}$ are a $C^1$ embedding into $R^{n+1-k}$ and an injection, respectively}.\vskip2mm\noindent
{\bf Lemma 3.4 of \cite{S6}.} {\it Assume (**) is already satisfied for any small linear perturbation of $p_1,...,p_{n-1}$.
Then we can choose such $p_1,...,p_{n-1}$ and $\{A_j\}_j$ so that (**) and (***) are satisfied for $k=2,...,n$.}\vskip2mm
 The actual statement of Lemma 3.4 of \cite{S6} is a little different to this.
We translate it so that we can apply it in the coming arguments.
As the proof becomes easier by the assumption in the above lemma, we do not prove.\vskip2mm\noindent
{\bf Proposition 3.6 of \cite{S6}.} {\it Triangulation theorem of definable $C^0$ functions (1) holds under (**) and (***).}\vskip2mm
 We use not only this proposition but also its proof.
In the proof we explicitly constructed a definable homeomorphism $\pi$ such that $f\circ\pi$ is PL.
We need that method of construction.
\section{Compact case}
{\it Proof of Theorem 2 for a compact definable $C^0$ manifold.} Let $M$ be a compact definable $C^0$ manifold.
By Triangulation theorem of definable sets, $M$ is definably homeomorphic to a compact polyhedron.
Hence we can assume $M$ is a compact polyhedron.
Moreover, by Uniqueness theorem of definable triangulation, if $M$ is definably homeomorphic to another compact polyhedron then they are PL homeomorphic.
Therefore, it remains to prove that $M$ is a PL manifold, i.e.~the following statement.\par
 {\it For each $x\in M$ there exists a polyhedral neighborhood $U_0$ of $x$ in $M$ such that $(U_0,x)$ is PL homeomorphic to $(\sigma_0,0)$, where $\sigma_0$ is a simplex such that $0\in\Int\sigma_0$.}\par
 By definition of a definable $C^0$ manifold there exists a compact definable neighborhood $U$ of $x$ in $M$ such that $(U,x)$ is definably homeomorphic to $(\sigma_0,0)$.
Apply Triangulation theorem of definable sets to $U$ and a simplicial decomposition of $M$ such that $x$ is a vertex.
Then there is a definable homeomorphism $\tau$ of $M$ such that $\tau(x)=x$ and $\tau(U)$ is a polyhedron.
Since $\tau(U)$ is a compact polyhedral neighborhood of $x$ in $M$, replacing $U$ with $\tau(U)$ we assume from the beginning that $U$ is a compact polyhedral neighborhood.
Then by Uniqueness theorem of definable triangulation, $(U,x)$ is PL homeomorphic to $(\sigma_0,0)$.\qed\vskip2mm\noindent
{\it Proof of Theorem 1 for a compact tame $C^0$ manifold.} Let $M$ be a compact tame $C^0$ manifold.
Let $\{S_n\}$ denote the minimal o-minimal structure where $M$ is definable.
Then by Triangulation theorem of definable sets, $M$ is $\{S_n\}$-definably homeomorphic to a compact polyhedron $X$.
Here $M$ and $X$ are definably homeomorphic in any o-minimal structure including $\{S_n\}$.
Thus, if $M$ is tamely homeomorphic to some compact polyhedron $X'$ then $M$ is definably homeomorphic to $X'$ in some o-minimal structure $\{S'_n\}$, and $M$ and $X$ are $\{S'_n\}$-definably homeomorphic since $\{S'_n\}\supset\{S_n\}$.
Therefore, $X$ and $X'$ are PL homeomorphic by Uniqueness theorem of definable triangulation.\par
 It remains to see that $X$ is a PL manifold.
By the above arguments it suffices to show that each point $x$ of $M$ has a neighborhood $U$ such that $(U,x)$ is tamely homeomorphic to $(\sigma_0,0)$, where $\sigma_0$ is a simplex such that $0\in\Int\sigma$.
This is clear by definition of a tame $C^0$ manifold.\qed
\section{Non-compact case}
Let $M\subset R^n$ be a non-compact definable $C^0$ manifold in Theorem 2.
Here we can assume $M$ is bounded in $R^n$ since $R^n$ is semialgebraically homeomorphic to $\{x\in R^n:|x|<1\}$.
Apply Triangulation theorem of definable sets to $(\overline M,\overline M-M)$.
Then $M$ is definably homeomorphic to a definable polyhedron $X$ such that $\overline X$ is a compact polyhedron.
Moreover, we see that $X$ is a PL manifold by Uniqueness theorem of definable triangulation, which we call informally a {\it triangulation} of $M$.
Hence we need to find a triangulation $X$ such that $\overline X$ is a compact PL manifold with boundary.
For this let us consider unique triangulations of general non-compact definable sets.
Indeed, uniqueness of triangulations of general compact definable sets was the key of the proof in the compact case, although all the arguments in this paper become simple if the sets are locally closed in their ambient spaces, i.e.\ the sets are open in their closures.
The notion of semilinear is convenient for this.
However, the category of all semilinear sets and semilinear $C^0$ maps is too large.
There are three typical examples of semilinear sets $X_1$ and $X_2$ which are definably homeomorphic but not semilinearly homeomorphic.
First, $X_1=R$ and $X_2=(0,\,1)$, secondly, $X_1=\partial\sigma-\{a\}$ and $X_2=(0,\,1)$, and, thirdly, $X_1=\sigma-\{0\}$ and $X_2=\sigma-\sigma/2$, where $\sigma$ is a 2-simplex such that $0\in\Int\sigma$ and $a\in\partial\sigma$.
We want to treat a natural family of semilinear sets where two semilinear sets are semilinearly homeomorphic if they are definably homeomorphic.
We avoid $X_1$ of the examples and consider only the following family which includes $X_2$ and any compact polyhedron.
We call a semilinear set $X$ in $R^n$ {\it standard} if $X$ is bounded in $R^n$ and there is a {\it collar} on $\overline X-X$ in $\overline X$, i.e.\ there exists a semilinear $C^0$ embedding $\phi:(\overline X-X)\times[0,\,1]\to\overline X$ such that $\phi(\cdot,0)=\id$ and $\phi((\overline X-X)\times[0,\,1))$ is an open neighborhood of $\overline X-X$ in $\overline X$ (Fig.\ 1 (a)).
We also call the image of $\phi$ a {\it collar} when we know $\phi$ in the context.
(In PL topology a collar is defined on a closed subpolyhedron.) 
Note that $\phi((\overline X-X)\times(0,\,1])\subset X$, the image of $\phi$ is not a triangle in Fig.\ 1 (a) and a non-compact PL manifold $X$ in $R^n$ is a standard semilinear set if and only if $\overline X$ is a compact PL manifold with boundary.
For a cell complex $K$, let $K'$ always denote the barycentric subdivision of $K$ in this paper (Fig.\ 1 (b)).
The following remark shows how standard semilinear sets are unified.\vskip2mm
\begin{figure}[h]
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\caption{}
\end{figure}\noindent
{\bf Remark 3.}
{\it Let $X$ be a standard semilinear set in $R^n$.\par\noindent
{\rm (i)} A semilinear $C^0$ function on $X$ is extensible to a PL function on $\overline X$.\par\noindent
{\rm (ii)} Set $X(0)=X,\ X(j)=\overline{X(j-1)}-X(j-1)\ (j=1,...).$
Then $\overline{X(j)}$ is the disjoint union of $X(j)$ and $X(j+1)$, $X(j)\supset X(j+2)$, $X(j)-X(j+2)$ is the inner point set of $X(j)$ in $\overline{X(j)}$, $X(j)$ is not necessarily standard (e.g.\ $X$ is the union of an open 2-simplex and a point of its boundary), and $X(2j)$ is standard.\par\noindent
{\rm (iii)} It is known that there are two compact PL manifolds with boundary which are not PL homeomorphic but whose interiors are PL homeomorphic \cite{Ma}.
However, two compact PL manifolds with boundary are PL homeomorphic if their interiors are semilinearly homeomorphic.
Moreover, a semilinear homeomorphism $f:X\to Y$ between standard semilinear sets is extended to a PL homeomorphism $\overline f:\overline X\to\overline Y$.
The last statement is not the case unless $X$ and $Y$ are standard.
A counter-example is $X=[0,\,1)\cup(1,\,2]$ and $Y=[0,\,1)\cup(2,\,3]$.}\vskip2mm\noindent
{\it Proof of Remark 3.} (i) Let $g$ be a semilinear $C^0$ function on $X$ and let $\phi:(\overline X-X)\times[0,\,1]\to\overline X$ be a collar on $\overline X-X$ in $\overline X$.
Then it suffices to see that $g\circ\phi|_{(\overline X-X)\times(0,\,1]}$ is extensible to a PL function on $(\overline{\overline X-X})\times[0,\,1]$.
Let $K$ be a simplicial decomposition of $(\overline{\overline X-X})\times[0,\,1]$ such that $(\overline X-X)\times(0,\,1]$ is the union of some open simplexes in $K$ and $g\circ\phi$ is linear on each open simplex in $K$ included in $(\overline X-X)\times(0,\,1]$.
For such an open simplex $\Int\sigma$, $g\circ\phi|_{\Int\sigma}$ is extended to a linear function on $\sigma$.
We need to show the uniqueness of these extensions.
For this it suffices to see the following statement.\vskip2mm\noindent
{\it Let $x_0\in\overline X-X$ and let $l_1$ and $l_2$ be distinct segments in $(\overline X-X)\times[0,\,1]$ with one end $(x_0,0)$ such that $l_1-\{(x_0,0)\}$ and $l_2-\{(x_0,0)\}$ are included in $(\overline X-X)\times(0,\,1]$.
Then $\lim_{l_1\ni(x,t)\to(x_0,0)}g\circ\phi(x,t)=\lim_{l_2\ni(x,t)\to(x_0,0)}g\circ\phi(x,t)$.}\par
 This is easy to see.
Let $\sigma_i\ (i=1,2)$ denote the 2-simplexes with two 1-faces $l_i$ and $\{x_0\}\times[0,\,1]$ if $l_i\not\subset\{x_0\}\times[0,\,1]$, and set $\sigma_i=l_i$ otherwise.
Then $\sigma_i-\{(x_0,0)\}\subset(\overline X-X)\times(0,\,1]$ and we see as above that $g\circ\phi|_{\sigma_i-\{(x_0,0)\}}$ are extended to PL functions on $\sigma_i$.
Hence 
$$
\lim_{l_1\ni(x,t)\to(x_0,0)}g\circ\phi(x,t)=\lim_{t\to 0}g\circ\phi(x_0,t)=\lim_{l_2\ni(x,t)\to(x_0,0)}g\circ\phi(x,t).
$$\par
 (ii) Only the last statement is not obvious.
By induction it suffices to show that $X(2)$ is standard.
Let $\phi$ be a collar as above.
We will see that $\phi|_{X(3)\times[0,\,t_0]}$ is a collar on $X(3)$ in $\overline{X(2)}$ for some $t_0\in(0,\,1]$, i.e.\ $\phi^{-1}(\overline{X(2)})\cap(X(1)\times[0,\,t_0])=X(3)\times[0,\,t_0]$.
The last equality is equivalent to the following statement.\vskip2mm\noindent
{\it Let $x\in X(1)$.
Then $\phi(\{x\}\times(0,\,t_0])\subset X-X(2)$ if and only if $x\in X(1)-X(3)$.}\par
 Note that the image of a compact polyhedron under a semilinear $C^0$ map is compact.
Assume that $x\in X(1)-X(3)$.
Then $x$ is an inner point of $X(1)$ in $\overline{X(1)}$, $X(1)\cap U$ is compact for some compact polyhedral neighborhood $U$ of $x$ in $R^n$, $\phi((X(1)\cap U)\times[0,\,1])$ is a compact neighborhood of $\phi(x,t)$ in $X$ for each $t\in(0,\,1)$ and hence $\phi(x,t)\not\in X(2)$.
Conversely assume that $\phi(x,t)\in X-X(2)$ for some $t\in(0,\,1)$.
Then there is a closed polyhedral neighborhood $U$ of $(x,t)$ in $X(1)\times[0,\,1]$ such that $\phi(U)$ is compact.
Such a $U$ is compact because $\phi^{-1}|_{\phi(U)}$ is a semilinear $C^0$ map.
Hence $x\in X(1)-X(3)$.
Thus the statement is proved.\par
 (iii) We prove the last statement.
By (i), $f$ and $f^{-1}$ are extended to PL maps $\overline f:\overline X\to\overline Y$ and $\overline{f^{-1}}:\overline Y\to\overline X$, respectively, and $\overline{f^{-1}}\circ\overline f:\overline X\to\overline X$ and $\overline f\circ\overline{f^{-1}}:\overline Y\to\overline Y$ are the $C^0$ extensions of the identity maps $f^{-1}\circ f:X\to X$ and $f\circ f^{-1}:Y\to Y$, respectively.
Hence $\overline f:\overline X\to\overline Y$ is a PL homeomorphism.\qed
\vskip2mm
 We need to consider plural semilinear sets at once.
We call a family of semilinear sets $(X;X_i)$ {\it standard} if $\{X_i\}$ is finite, $X$ is standard and we can choose the semilinear $C^0$ embedding $\phi:(\overline X-X)\times[0,\,1]\to\overline X$ so that $\phi^{-1}(X_i)=X'_i\times(0,\,1]$ for each $i$ and some semilinear subset $X'_i$ of $\overline X-X$.
We call $\phi$ or $\Ima\phi$ a {\it collar} on $\overline X-X$ in $(\overline X;X,X_i)$.
Remark 3 (ii) is generalized so that $(X(2j);X(2j)\cap X_i)$ is standard for each $j$.
Note that for non-compact, semilinear and bounded $X$ and a simplicial decomposition $K$ of $\overline X$, $(X;\sigma\cap X:\sigma\in K)$ is not standard if $X$ is of dimension $>1$ locally at $\overline X-X$.
Now we state uniqueness triangulations of non-compact definable sets.\vskip2mm\noindent
{\bf Theorem 4. (Unique standard triangulation theorem of definable sets)}\\
(i) {\it A finite family of definable sets $(X;X_i)$ is definably homeomorphic to a unique standard family of semilinear sets when $X$ is not compact.}\\
(ii) {\it Moreover, a definable homeomorphism between standard families of semilinear sets is definably isotopic to a semilinear homeomorphism through homeomorphisms.}\vskip2mm
Theorem 2 and Theorem 1 in the non-compact case follow quickly from Theorem 4 as in the compact case.\vskip2mm\noindent
{\it Proof of Theorem 2 for a non-compact definable $C^0$ manifold.} Let $M$ be a non-compact definable $C^0$ manifold.
By Theorem 4, $M$ is definably homeomorphic to a standard semilinear set $X$.
Then $X$ is a PL manifold by the same reason as in the proof of Theorem 2 in the compact case, and $\overline X$ is a compact PL manifold with boundary as we already noted.
It remains to see uniqueness of $\overline X$.
Let $M$ be definably homeomorphic to the interior of another compact PL manifold with boundary $M_1$.
Then $\Int M_1$ is standard.
Hence by uniqueness in Theorem 4, $\overline X$ and $M_1$ are PL homeomorphic.\qed\vskip2mm\noindent
{\it Proof of Theorem 1 for a non-compact tame $C^0$ manifold.} We proceed as in the compact case.
Let $M$ be a non-compact tame $C^0$ manifold.
Let $\{S_n\}$ denote the minimal o-minimal structure where $M$ is definable.
Then by Theorem 4, $M$ is $\{S_n\}$-definably homeomorphic to a standard semilinear set $X$, and $M$ and $X$ are definably homeomorphic in any o-minimal structure including $\{S_n\}$.
Thus, if $M$ is tamely homeomorphic to some standard semilinear set $X'$ then $M$ is definably homeomorphic to $X'$ in some o-minimal structure $\{S'_n\}$, and $M$ and $X$ are $\{S'_n\}$-definably homeomorphic.
Therefore, $X$ and $X'$ are semilinearly homeomorphic by Theorem 4.
It follows that $\overline X$ and $\overline{X'}$ are PL homeomorphic.
Hence it remains only to see that $\overline X$ is a PL manifold with boundary, i.e.~$X$ is a PL manifold.
However, we have already shown this in the proof of the compact case.\qed
\section{Lemmas}
From now on we are engaged only in proving Theorem 4.
We will use the following lemmas in its proof.\vskip2mm\noindent
{\bf Lemma 5.} {\it Let $K$ be a cell complex generated by one cell $\sigma_0$, let $\sigma_1$ be a proper face of $\sigma_0$, and let $\sigma_2,...,\sigma_k$ be some proper faces of $\sigma_1$.
Set $\delta=\cup_{j=1}^k\Int\sigma_j$.
Then a definable isotopy of $\partial\sigma_0-\delta$ preserving $\{\sigma-\delta:\sigma\in K|_{\partial\sigma_0}\}$ is extensible to a definable isotopy of $\sigma_0-\delta$.}\vskip2mm\noindent
{\it Proof of Lemma 5.} 
We proceed by induction on the dimension of $\delta$.
Let $\partial\alpha_t,\ 0\le t\le 1$, denote the definable isotopy of $\partial\sigma_0-\delta$.
Assume that $0\in\Int\sigma_0$.
Then it is natural to define an extension $\alpha_t$ by $\alpha_t(sy)=s\partial\alpha_t(y)$ for $(y,s,t)\in(\partial\sigma_0-\delta)\times[0,\,1]^2$.
However, such an $\alpha_t$ is not defined on the interior of $\sigma_0$.
We need to modify the definition.\par
 First we reduce the problem to the case where $\partial\alpha_t=\id$ on $\partial\sigma_1-\delta$ (not $\partial\sigma_0-\delta$).
There is a cellular subdivision $K_1$ of $K$ such that $K|_{\partial\sigma_1}=K_1|_{\partial\sigma_1}$ and for each $\sigma$ in $K_1$ not included in $\partial\sigma_1$, $(\sigma;\sigma\cap\sigma_j:j=2,...,k)$ satisfies the conditions on $(\sigma_0;\sigma_j:j=1,...,k)$ in Lemma 5.
Indeed, choose one point in each open cell in $K$ not included in $\partial\sigma_1$, and define a cell complex $K_1$ such that $K|_{\partial\sigma_1}=K_1|_{\partial\sigma_1}$ and $K_1^0$ consists of the points and $K^0\cap\partial\sigma_1$ in the same way as we have defined a derived subdivision of a cell complex.
Then $K_1$ satisfies the conditions on $(\sigma_0;\sigma_j:j=1,...,k)$.
Let us consider $\partial\alpha_t|_{\partial\sigma_1-\delta}$ and $K_1$.
Then by using the induction hypothesis we can extend $\partial\alpha_t|_{\partial\sigma_1-\delta}$ to a definable isotopy $\beta_t,\ 0\le t\le1$, of $\sigma_0-(\delta-\Int\sigma_1)$ preserving $\{\sigma-(\delta-\Int\sigma_1):\sigma\in K_1\}$.
Set $\partial\gamma_t=\beta_t^{-1}\circ\partial\alpha_t$, which is a definable isotopy of $\partial\sigma_0-\delta$ preserving $\{\sigma-\delta:\sigma\in K|_{\partial\sigma_0}\}$ and whose restriction to $\partial\sigma_1-\delta$ is the identity map.
If $\partial\gamma_t$ is extended to a definable isotopy $\gamma_t$ of $\sigma_0-\delta$, then $\beta_t\circ\gamma_t$ is a definable isotopy of $\sigma_0-\delta$ and is an extension of $\partial\alpha_t$.
Hence we can assume that $\partial\alpha_t=\id$ on $\partial\sigma_1-\delta$ from the beginning.\par
 Let $\xi:\partial\sigma_0\times[0,\,1]\to\sigma_0$ be a semialgebraic $C^0$ map such that $\xi(x,s)=x$ for $(x,s)\in(\partial\sigma_0\times\{0\})\cup(\sigma_1\times[0,\,1])$ and $\xi|_{(\partial\sigma_0-\sigma_1)\times[0,\,1]}$ is a $C^0$ embedding.
 For example, let $\xi$ be the composite of a map $\partial\sigma_0\times[0,\,1]\ni(x,s)\to(x,s\rho(x))\in\partial\sigma_0\times[0,\,1]$ and a collar of $\partial\sigma_0$ in $\sigma_0$, where $\rho:\partial\sigma_0\to[0,\,1]$ is a PL map with zero set $\sigma_1$.
Let us define isotopies $\tilde\alpha_t$, $0\le t\le1$, of $(\partial\sigma_0-\delta)\times[0,\,1]$ to be $\tilde\alpha_t(x,s)=(\partial\alpha_{t(1-s)}(x),s)$, and $\alpha_t$ of $\Ima\xi-\delta$ by 
$$
\alpha_t(x)=\xi\circ\tilde\alpha_t\circ\xi^{-1}(x)\quad\text{for}\ x\in\Ima\xi.
$$
Then $\tilde\alpha_t(x,0)=(\partial\alpha_t(x),0)$, $\tilde\alpha_t=\id$ on $(\partial\sigma_0-\delta)\times\{1\}$, $\alpha_t$ is a well-defined definable isotopy of $\Ima\xi-\delta$ since $\partial\alpha_t=\id$ on $\partial\sigma_1-\delta$, and $\alpha_t=\id$ on $\xi(\partial\sigma_0\times\{1\})-\delta$---the boundary of $\Ima\xi-\delta$ in $\sigma_0-\delta$.
Hence we can extend $\alpha_t$ to a definable isotopy of $\sigma_0-\delta$ by setting $\alpha_t=\id$ outside of the image of $\xi$.
Thus Lemma 5 is proved.\qed\vskip2mm\noindent
{\bf Lemma 6. (Lifts of definable homeomorphisms)}
{\it Let $\gamma:L\to M$ be a cellular map between finite cell complexes and $D$ a subset of $|M|$.
Let $\delta_M$ be a definable homeomorphism of $|M|$ preserving $M$ such that $\delta_M=\id$ on $D$.
Then there exists a definable homeomorphism $\delta_L$ of $|L|$ preserving $L$ such that $\delta_L=\id$ on $\gamma^{-1}(D)$ and $\delta_M\circ\gamma=\gamma\circ\delta_L$.}\vskip2mm\noindent
{\it Proof of Lemma 6.} 
For each $\sigma\in L$ we will construct a definable homeomorphism $\delta_\sigma$ of $\sigma$ such that $\delta_M\circ\gamma=\gamma\circ\delta_\sigma$ on $\sigma$ and $\delta_\sigma=\delta_{\sigma'}$ on $\sigma\cap\sigma'$ for other $\sigma'\in L$.
If $\gamma|_\sigma$ is injective, then $\delta_\sigma$ is uniquely determined.
In the other case, we define $\delta_\sigma$ by a cross-section as follows.
Let $c_\sigma:\gamma(\sigma)\to\sigma$ be a PL cross-section of $\gamma|_\sigma$ such that $c_\sigma(x)\in\Int(\gamma|_\sigma)^{-1}(x)$ for each $x\in\gamma(\sigma)$ and $c_\sigma=c_{\sigma'}$ on $\gamma(\sigma')$ for each face $\sigma'$ of $\sigma$ with $\sigma'=(\gamma|_\sigma)^{-1}(\gamma(\sigma'))$, which we can construct by double induction on $\dim\sigma-\dim\gamma(\sigma)$ and $\dim\sigma$ by the Alexander trick, (8) of Sect.~2.
Note $(\gamma|_\sigma)^{-1}(x)=c_\sigma(x)*\partial((\gamma|_\sigma)^{-1}(x))$.
By induction on the dimension of $\sigma$ we assume that $\delta_\sigma$ is already constructed on the boundary of $\sigma$.
Let $\delta_{\partial\sigma}$ denote the homeomorphism of the boundary of $\sigma$.
We naturally extend the restriction of $\delta_{\partial\sigma}$ to $\cup_{x\in\gamma(\sigma)}\partial((\gamma|_\sigma)^{-1}(x))$ (not $\delta_{\partial\sigma}$ itself) to $\delta_\sigma$ by 
$$
\delta_\sigma\big(ty+(1-t)c_\sigma(x)\big)=t\delta_{\partial\sigma}(y)+(1-t)c_\sigma\circ\delta_M(x)\quad\text{for}\ (y,t)\in\partial((\gamma|_\sigma)^{-1}(x))\times[0,\,1].
$$
Then $\delta_\sigma$ is a definable homeomorphism of $\sigma$ preserving $L|_\sigma$, $\gamma\circ\delta_\sigma=\delta_M\circ\gamma$ on $\sigma$, moreover, $\delta_\sigma=\delta_{\partial\sigma}$ on $\partial\sigma-\cup_{x\in\gamma(\sigma)}\partial((\gamma|_\sigma)^{-1}(x))$ because of the method of construction of $\delta_\sigma$, hence $\delta_\sigma$ is an extension of $\delta_{\partial\sigma}$ and $\delta_\sigma=\id$ on $\sigma\cap\gamma^{-1}(D)$.
Thus Lemma 6 is proved.\qed\vskip2mm\noindent
{\bf Lemma 7. (Local triangulations of definable $C^0$ maps)}
{\it Let $K$ be a finite simplicial complex in $R^n$ with underlying polyhedron $X$ and let $H=(H_1,H_2):X\to R^2$ be a definable $C^0$ map such that $H_1\ge 0,\ H_2\ge 0$ and the zero sets of $H_1$ and $H_2$ are the underlying polyhedra of some subcomplexes of $K$.
Then there exist a definable isotopy $\zeta_t,\ 0\le t\le 1$, of $X$ preserving $K$ and a definable neighborhood $N$ of $(0,\,\epsilon]\times\{0\}$ in $(0,\,\epsilon]\times R$ for some positive $\epsilon\in R$ such that the $H\circ\zeta_1|_{H^{-1}(\overline N)}$ is extensible to a PL map $\tilde H=(\tilde H_1,\tilde H_2):X\longrightarrow R^2$ satisfying $\tilde H_1\ge0,\ \tilde H_2\ge0,\ \tilde H_1^{-1}(0)=H_1^{-1}(0)$ and $\tilde H_2^{-1}(0)=H_2^{-1}(0)$.\par
 If $H_1$ is PL from the beginning, then we can choose $\zeta_t$ so that $H_1\circ\zeta_1=H_1$ on $H^{-1}(\overline N)$.}\vskip2mm
 Note that a triangulation of $H$ is impossible in general, e.g.\ the blowing-up $[0,\,1]^2\ni(x_1,x_2)\allowbreak\to(x_1,x_1x_2)\in R^2$ does not admit a triangulation.\vskip2mm\noindent
{\it Proof of Lemma 7.} 
If $\Ima H\cap N$ is of dimension $\le1$ for some $N$ in Lemma 7 (i.e.\ $\Ima H\cap N$ is the empty set or $(0,\,\epsilon]\times\{0\}$), then Lemma 7 is obvious or follows from Triangulation theorem of definable $C^0$ functions (1).
Hence we assume that $\Ima H\cap N$ is of dimension $2$ for any $N$.
Let us consider the former half of Lemma 7.
We only construct a definable homeomorphism $\zeta$ of $X$ preserving $K$ such that the $H\circ\zeta|_{H^{-1}(\overline N)}$ is extensible to a PL map because the other condition that $\zeta$ is the finishing homeomorphism of some isotopy can be clearly satisfied by the usual arguments.
We use the same argument as in the proof of Triangulation theorem of definable $C^0$ functions (1).
Set $A=\graph H$, and let $R^n\times R^2\stackrel{p_1}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}R^2$ denote the projections ignoring the respective first factors.
If there is a sequence of definable $C^1$ stratifications $\{A''_{j''}\}\stackrel{p_1}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}\{p_n\circ\cdots\circ p_1(A''_{j''})\}$ such that $\{A''_{j''}\}$ is a stratification of $A$ compatible with $\{A\cap(\sigma\times R^2):\sigma\in K\}$, Conditions (**) and (***) in Sect.\ 2 are satisfied and $\{p_n\circ\cdots\circ p_1(A''_{j''})\}$ are the open simplexes in some simplicial complex in $R^2$, then by Proposition 3.6 and its proof there exists a definable homeomorphism $\tau$ of $X$ preserving $K$ such that $H\circ\tau$ is PL.
We will obtain such a sequence of definable $C^1$ stratifications by shrinking $A$ to $A\cap(R^n\times\overline N)$.\par
 Let $p:R^n\times R^2\to R^2$ denote the projection and $p_{n+1}:R^2\to R$ the projection to the first (not last) factor.
Set $B_{(u,v)}=\{x\in R^n:(x,u,v)\in B\}$ for $(u,v)\in R^2$ and any subset $B$ of $R^n\times R^2$.
Let $p|_A:\{A_j\}_j\to\{ p(A_j)\}_j$ be a definable $C^1$ stratification of $p|_A$ compatible with $(\{A\cap(\sigma\times R^2):\sigma\in K\},\emptyset)$.
Let $A'$ denote the union of the $A_j$ such that $A_{j(u,v)}$ is of dimension smaller than $n$ for each $(u,v)\in R^2$.
Set $S^{n-1}=\{\lambda\in R^n:|\lambda|=1\}$, let $T_{(u,v)}\ (\subset S^{n-1})$ denote the closure of the set of singular directions for $A'_{(u,v)}$ and set $T=\{(\lambda,u,v)\in S^{n-1}\times R^2:\lambda\in T_{(u,v)}\}$.
Then $T$ is definable and $T_{(u,v)}$ is a definable set of dimension smaller than $n-1$ for each $(u,v)$.
Note that there is the case where $\cup_{(u,v)\in R^2}T_{(u,v)}=S^{n-1}$.
We need $N$ such that $\cup_{(u,v)\in\overline N}T_{(u,v)}\not=S^{n-1}$, i.e.\ there is a non-singular direction for $A'_{(u,v)}$ for any $(u,v)\in\overline N$.\par
We will find a closed definable subset of $S^{n-1}$ of dimension smaller than $n-1$ any definable neighborhood of which includes $\cup_{(u,v)\in\overline N}T_{(u,v)}$ for some $N$.
As the set we will choose $\cap_{u_0>0}\cup_{0\le u\le u_0}\overline{\cup_{0\le v\le h(u)}T_{(u,v)}}$ for some small non-negative definable $C^0$ function $h$ on $R$ with zero set $\{0\}$.
The set should includes $T_{(0,0)}$.
If $(\overline T)_{(0,0)}$ is of dimension smaller than $n-1$, then it satisfies the requirement.
However, it may be of dimension $n-1$ since the parameter of $T_{(u,v)}$ is of two variables.
(For example, the set $C=\{(x,u,v)\in [0,\,1]^3:x u=v\}$ has the property\,: $C=\overline{\cup_{(u,v)\not=(0,0)}C_{(u,v)}\times\{(u,v)\}},\ \dim C_{(u,v)}=0$ for $(u,v)\not=(0,0)$ with $C_{(u,v)}\not=\emptyset$, and $\dim C_{(0,0)}=1$.
If $C$ is of one parameter, $C=\cup_{u\in R}C_u\times\{u\}$, $C_0\not=\emptyset$ and $C=\overline{\cup_{u\not=0}C_u\times\{u\}}$, then $\dim C_0=\dim C_u$ and $C_u\not=\emptyset$ for some $u\not=0$ arbitrarily near 0.) 
Hence we choose another set smaller than $(\overline T)_{(0,0)}$ by reducing the problem to the case of one variable.
Set 
$$
U=\cup_{u\in[0,\,1]}(T\cap(S^{n-1}\times\{u\}\times R)\overline).
$$
Then $T\subset U$, $U$ is a definable subset of $S^{n-1}\times R^2$ and $U_{(u,v)}$ is closed and of dimension smaller than $n-1$ for each $(u,v)$.
However, $U_{(0,0)}$ does not necessarily satisfies the requirement.
We need to enlarge $U_{(0,0)}$ this time.
Next, we set 
$$
V=\cup_{v\in[0,\,1]}(U\cap(S^{n-1}\times R\times\{v\})\overline).
$$
Then $U\subset V$, $V$ is a definable subset of $S^{n-1}\times R^2$, $V\cap(S^{n-1}\times R\times\{0\})$ is closed, $V_{(u,v)}$ is closed and of dimension smaller than $n-1$ for each $(u,v)$, and, moreover, $V_{(0,0)}$ fulfills the requirement as follows.\par
 Let $Q$ be any definable neighborhood of $V_{(0,0)}$ in $S^{n-1}$.
We need to see that the set $N\overset{\text{def}}{=}\{(u,v)\in(0,\,\epsilon]\times R:Q\supset V_{(u,v)}\}$ is a neighborhood of $(0,\,\epsilon]\times\{0\}$ in $(0,\,\epsilon]\times R$ for some $\epsilon\in(0,\,1]$.
There exists $\epsilon\in(0,\,1]$ such that $Q\times[0,\,\epsilon]\times\{0\}$ is a neighborhood of $V\cap(S^{n-1}\times[0,\,\epsilon]\times\{0\})$ in $S^{n-1}\times[0,\,\epsilon]\times\{0\}$ since $V\cap(S^{n-1}\times[0,\,\epsilon]\times\{0\})$ is closed.
Hence $N\cap(\{u\}\times R)$ is a neighborhood of $(u,0)$ in $\{u\}\times[0,\,\epsilon]$ for each $u\in[0,\,\epsilon]$.
Then, since $N$ is definable, there is a positive definable function $h$ on $(0,\,\epsilon]$ such that $N\supset\{(u,v)\in(0,\,\epsilon]\times R:0\le v\le h(u)\}$.
Shrink $\epsilon$ so that $h$ is of class $C^0$ on $(0,\,\epsilon]$.
Then $N$ is a neighborhood of $(0,\,\epsilon]\times\{0\}$ in $(0,\,\epsilon]\times R$.\par
 Thus we obtain $V_{(0,0)}$ and $N$ such that the $\cup_{(u,v)\in\overline N}T_{(u,v)}$ is included in a small definable neighborhood of $V_{(0,0)}$.
Then $\cup_{(u,v)\in\overline N}T_{(u,v)}\not= S^{n-1}$.
Hence there exists a non-singular direction for $A'_{(u,v)}$ for any $(u,v)\in\overline N$.
After linearly changing the coordinate system of $R^n$ we can assume that the element $(1,0,...,0)$ of $R^n$ or any element of $R^n$ near $(1,0,...,0)$ is not a singular direction for $A'_{(u,v)}$ for any $(u,v)\in\overline N$.
Let $\{N_k\}$ be a definable $C^1$ stratification of $\overline N$ and let $\{A'_{j'}\}\stackrel{p_1}{\longrightarrow}\{p_1(A'_{j'})\}$ be a definable $C^1$ stratification of $p_1|_{A\cap(R^n\times\overline N)}$ compatible with $(\{A_j\},\emptyset)$ such that $\{A'_{j'}\cap(R^n\times N_k)-A'\}=\{A_j\cap(R^n\times N_k)-A'\}$.
Then Condition (**) is satisfied.\par
 Repeat the same arguments for $\{p_1(A'_{j'})\}$ and $R^{n-1}\times R\stackrel{p_2}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}R^2$.
Then shrinking $N$ we obtain a sequence of definable $C^1$ stratifications $\{A''_{j''}\}\stackrel{p_1}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}\{p_n\circ\cdots\circ p_1(A''_{j''})\}$ such that $\{A''_{j''}\}$ is a stratification of $A\cap(R^n\times\overline N)$ compatible with $\{A\cap(\sigma\times\overline N):\sigma\in K\}$ and Condition (**) is satisfied.
Moreover, this holds after any small perturbation of $p_1,...,p_{n-1}$.
Here we can assume by Lemma 3.4 that Condition (***) is also satisfied.
However, the remaining condition that $\{p_n\circ\cdots\circ p_1(A''_{j''})\}$ is the set of the open simplexes in some simplicial complex in $R^2$ cannot be satisfied because of $\cup_{j''}p_n\circ\cdots\circ p_1(A''_{j''})=\overline N\cap\Ima H$.
We solve this problem as follows.\par
 Shrinking $N$ we can assume that $\overline N\subset\Ima H$, $\overline N=\{(u,v)\in[0,\,\epsilon]\times R:0\le v\le h(u)\}$ and $\{p_n\circ\cdots\circ p_1(A''_{j''})\}$ is the natural stratification of $\overline N$, i.e.\ $\{p_n\circ\cdots\circ p_1(A''_{j''})\}=\{(0,0),\,(\epsilon,0),\,(\epsilon,h(\epsilon)),\,\allowbreak\graph h|_{(0,\,\epsilon)},\,(0,\,\epsilon)\times\{0\},\,\{\epsilon\}\times(0,\,h(\epsilon)),\,\{(u,v)\in(0,\,\epsilon)\times R:0<v<h(u)\}\}$, where $h$ is a non-negative definable $C^1$ function on $[0,\,\epsilon]$ with zero set $\{0\}$.
Let $\sigma_N$ denote the 2-simplex in $R^2$ with vertices $(0,0),\,(\epsilon,0)$ and $(\epsilon,h(\epsilon))$, and let $\beta:\overline N\to\sigma_N$ be a definable homeomorphism fixing $[0,\,\epsilon]\times\{0\}$ and carrying each segment $l$ in $\overline N$ parallel to $\{0\}\times R$ to the segment $l'$ in $\sigma_N$ such that $l$ and $l'$ have the same extension to a line.
Set $\tilde\beta(x,u,v)=(x,\beta(u,v))$ for $(x,u,v)\in R^n\times\overline N$.
Let us consider $\beta\circ H|_{H^{-1}(\overline N)},\,\tilde\beta(A\cap(R^n\times\overline N))$ and its definable $C^1$ stratification $\{\tilde\beta(A''_{j''})\}$ in place of $H|_{H^{-1}(\overline N)},\,A\cap(R^n\times\overline N)$ and $\{A''_{j''}\}$.
Then $\tilde\beta(A\cap(R^n\times\overline N))$ is the graph of $\beta\circ H|_{H^{-1}(\overline N)}$, and the sequence $\{\tilde\beta(A''_{j''})\}\stackrel{p_1}{\longrightarrow}\cdots\stackrel{p_n}{\longrightarrow}\{p_n\circ\cdots\circ p_1\circ\tilde\beta(A''_{j''})\}$ satisfies the remaining condition since $\{p_n\circ\cdots\circ p_1\circ\tilde\beta(A''_{j''})\}$ is the set of open simplexes in the simplicial complex generated by $\sigma_N$.\par
 There is another problem when we apply Proposition 3.6.
Namely, $H^{-1}(\overline N)$ does not necessarily coincide with $X$.
To take this into account, the proof of Proposition 3.6 says that there exist a compact subpolyhedron $X_1$ of $X$ and a definable homeomorphism $\tau:X_1\to H^{-1}(\overline N)$ such that $\tau(\sigma\cap X_1)\subset\sigma$ for each $\sigma\in K$, $\beta\circ H\circ\tau$ is PL and $\tau$ is extended to a definable homeomorphism $\tau$ of $X$ preserving $K$.
Shrink $N$ to another $N_1$ so that $H^{-1}(\overline{N_1})\subset X_1$, which is possible because $\tau$ preserves the inverse images of $(0,0)$ and $R\times\{0\}$ under $H$ and hence $X_1$ is a neighborhood of $H^{-1}((0,\,\epsilon]\times\{0\})$ for shrunk $\epsilon$.
Then $\beta\circ H\circ\tau|_{H^{-1}(\overline{N_1})}$ is extensible to a PL map $\tilde H=(\tilde H_1,\tilde H_2):X\to R^2$.\par
 It remains to find a definable homeomorphism $\zeta$ of $X$ preserving $K$ such that $\beta\circ H\circ\tau=H\circ\zeta$ on $H^{-1}(\overline{N_2})$ for some smaller $N_2$.
Let $\beta\circ H\circ\tau|_{X_1}:L\to M$ be a simplicial decomposition of $\beta\circ H\circ\tau|_{X_1}:X_1\to\sigma_N$ such that each simplex in $L$ is included in some simplex in $K$.
Let $\sigma_M$ be a small simplex in $R^2$ such that one of the vertices is the origin, another lies on $R\times\{0\}$ and $\sigma_M$ and $\beta(\sigma_M)$ are included in a simplex in the barycentric subdivision $M'$.
Then we can replace $N_1$ with $\beta^{-1}(\sigma_M)-\{0\}$, and by the Alexander trick we can extend $\beta|_{\sigma_M}$ to a definable homeomorphism $\beta_M$ of $\sigma_N$ preserving $M$.
Hence the existence of $\zeta$ follows from Lemma 6.
Moreover, the conditions $\tilde H_1\ge0,\ \tilde H_2\ge0,\ \tilde H_1^{-1}(0)=H_1^{-1}(0)$ and $\tilde H_2^{-1}(0)=H_2^{-1}(0)$ are obviously satisfied.
Thus the former half of Lemma 7 is proved.\par
 Let us consider the latter half of Lemma 7.
Apply (13) of Sect.\ 2 to PL functions $H_1$ and $\tilde H_1$.
Then there is a PL isotopy $\zeta'_t,\ 0\le t\le1$, of $X$ preserving $K$ such that $\tilde H_1\circ\zeta'_1=H_1$ on $H_1^{-1}([0,\epsilon'])$ for some sufficiently small $\epsilon'>0$.
Replace $\epsilon$ with $\epsilon'$, $N$ with $N\cap[0,\,\epsilon']\times R$, $\zeta_t$ with $\zeta_t\circ\zeta'_t$ and $\tilde H$ with $\tilde H\circ\zeta'_1$.
Then the latter half follows, and we complete the proof of Lemma 7.\qed

\section{Proof of Theorem 4}
{\it Proof of existence of a standard family in (i) of Theorem 4.}
By Triangulation theorem of definable sets we can assume that $X$ and $X_i$ are the unions of some open simplexes in some finite simplicial complex $K$ in $R^n$ with $|K|=\overline X$.
Let $v_\sigma$ denote the barycenter of a simplex $\sigma$.
For each $\sigma\in K$ we enlarge the slit $\Int\sigma$ in $\overline X$ to its neighborhood in $\overline X$ by a semialgebraic (not semilinear) $C^0$ embedding $\tau_\sigma:\overline X-\Int\sigma\to\overline X-\Int\sigma$ as follows.
We set $\tau_\sigma=\id$ outside of $\Int|\st(v_\sigma,K')|-\Int\sigma$.
In the case where $\sigma$ is a vertex, we define $\tau_\sigma$ inside so that the $\overline{X-\Ima\tau_\sigma}$ is $\sigma*|\lk(\sigma,K'')|$ and $\tau_\sigma$ linearly carries each (open) segment $l$ joining $\sigma$ and a point $x$ of $|\lk(\sigma,K')|$ to the segment $l-|\st(\sigma,K'')|$.
Here $\lk(\sigma,K'')$ denotes the link of $\sigma$ in $K''$.
(Note 
$$
\tau_\sigma((1-t)\sigma+t x)=(1-t)\sigma/2+(1+t)x/2\quad\text{for}\ (x,t)\in|\lk(\sigma,K')|\times(0,\,1].)\leqno{(*)}
$$
In the general case of $\sigma$ we define $\tau_\sigma$ on $v_\sigma*|\lk(\sigma,K)|-\{v_\sigma\}$ in the same way and naturally extend it to $|\st(\sigma,K)|-\Int\sigma$ by 
$$
\tau_\sigma(tx+(1-t)y)=tx+(1-t)\tau_\sigma(y)\quad\text{for}\ (x,y,t)\in\partial\sigma\times(v_\sigma*|\lk(\sigma,K)|-\{v_\sigma\})\times[0,\,1].
$$
(Note that any element of $|\st(\sigma,K)|$ is of the form $t x+(1-t)y,\ (x,y,t)\in\partial\sigma\times(v_\sigma*|\lk(\sigma,K)|)\times[0,\,1]$.) 
Then 
$$
\overline X-\overline{\Ima\tau_\sigma}=\{x\in\Int|\st(v,K'')|:v\in K^{\prime\prime0}\cap\Int\sigma\},
$$
$$
\Ima\tau_{\sigma_1}\circ\tau_{\sigma_2}=\Ima\tau_{\sigma_1}\cap\Ima\tau_{\sigma_2}\quad\text{for}\ \sigma_1\not=\sigma_2\in K\ \text{with}\ \dim\sigma_1\le\dim\sigma_2.
$$

\begin{figure}[h]
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\psdot(11.4,1.5)
\rput(10.66,1.3){$\sigma$}
\pscurve{->}(10.8,1.25)(11.1,1.22)(11.36,1.2)
\rput(10.65,1.68){$v_\sigma$}
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\rput(10,1.15){$v_l$}
\pscurve{->}(10.17,1.2)(10.5,1.39)(10.87,1.5)
\rput(10.7,0.71){$v_1$}
\pscurve{->}(10.85,0.7)(11.1,0.68)(11.35,0.69)
\rput(10.05,2.3){$|\!\st(\sigma,K)|$}
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\pscurve{->}(10.8,2.3)(11.4,2.4)(11.8,2.2)(11.9,1.9)
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\rput(13.37,1){$|\!\st(v_\sigma,K')|$}
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\pscurve{->}(12.5,1)(12.1,1.1)(11.04,1.39)
\rput(13.36,2){$\overline X\!-\!\Ima\tau_\sigma$}
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\pscurve{->}(12.53,1.86)(12.1,1.75)(11.48,1.59)
\end{pspicture}
\caption{}\end{figure}\vskip2mm\noindent
(Fig.\ 2 describes the image of $\tau_\sigma$, where $K$ is generated by two 2-simplexes with common 1-face $\sigma$.)\par
 Let $\{\sigma_1,...,\sigma_k\}=\{\sigma\in K:X\cap\Int\sigma=\emptyset\}$, order them so that $\dim\sigma_1\le\cdots\le\dim\sigma_k$ and set $\tau=\tau_{\sigma_1}\circ\cdots\circ\tau_{\sigma_k}$.
Then $\tau$ is a semialgebraic $C^0$ embedding of $(X;X_i)$ into $(X;X_i)$, $\overline{\Ima\tau}\subset X$, 
$$
\overline X-\overline{\Ima\tau}=\overline X-\cap_{j=1}^k\overline{\Ima\tau_{\sigma_j}}=\{x\in\Int|\st(v,K'')|:v\in K^{\prime\prime0}-X\}, 
$$
hence $\overline X-\overline{\Ima\tau}$ is a neighborhood of $\overline X-X$ in $\overline X$, $(\Ima\tau;\tau(X_i))$ is a family of semilinear sets and is semialgebraically homeomorphic to $(X;X_i)$, and it is, moreover, standard for the following reason.\par
 Set $V=\overline{\Ima\tau}-\Ima\tau$.
We define a collar on $V$ in $\overline{\Ima\tau}$ by modifying $\tau$.
Let the above $\tau_\sigma$ be rewritten as $\tau_{\sigma 1/2}$ and define $C^0$ embeddings $\tau_{\sigma t_0}:\overline X-\Int\sigma\to\overline X-\Int\sigma$ for $t_0\in(0,\,1]$ by replacing $(*)$ with 
$$
\tau_\sigma((1-t)\sigma+t x)=t_0(1-t)\sigma+(1+t_0 t-t_0)x.
$$
Let us define $\tau_{t_0}$ to be $\tau_{\sigma_1 t_0}\circ\cdots\circ\tau_{\sigma_k t_0}$ for $t_0\in(0,\,1]$.
Then $\tau_{t_0}$ are semialgebraic $C^0$ embeddings of $(X;X_i)$ into $(X;X_i)$, $\overline{\Ima\tau_{t_0}}\subset\Ima\tau_{t_0'}$ for $t_0<t_0'$, $\tau_1=\id$, the map $\pi:V\times[0,\,1]\ni(x,t)\to\tau_{1-t/2}(x)\in\overline{\Ima\tau_{1/2}}$ is a semialgebraic $C^0$ embedding, $\pi(\cdot,0)=\id$, the $\pi(V\times[0,\,1))\ (=\overline{\Ima\tau_{1/2}}-\overline{\Ima\tau_{1/4}})$ is an open semilinear neighborhood of $V$ in $\overline{\Ima\tau_{1/2}}$, and $\pi^{-1}(\tau_{1/2}(X_i))=(V\cap X_i)\times(0,\,1]$.
Thus $\pi$ satisfies the conditions on a collar except that $\pi$ is of semilinear class.
We need to modify $\pi$ to a semilinear embedding, say, $\xi:V\times[0,\,1]\to\overline{\Ima\tau_{1/2}}$.
It is easy in the case $\overline V=V$ and becomes apparent.
In the other case, there are two problems.
One is that for any convergent directed family of points $V\ni x_\alpha\to x\in\{x\in\sigma:\sigma\in K'',\,\sigma\subset\overline V-V\}$, the $\pi(\{x_\alpha\}\times[0,\,1])$ converges to the point $\{x\}$ but $\xi(\{x_\alpha\}\times[0,\,1])$ does not converge to a point as in Fig.\ 1 (a).
We solve this problem by shrinking $V\times[0,\,1]$---the domain of definition of $\xi$---to the following set $\Xi$ so that the $\xi(\Xi\cap(\{x_\alpha\}\times[0,\,1]))$ converges to a point.\par
 Note that $V$ is the union of some open simplexes in $K''$ and
$$
\pi^{-1}(\sigma\cap\overline{\Ima\tau_{1/2}})=(V\cap\sigma)\times[0,\,1]\quad\text{for}\ \sigma\in K'\ \text{with}\ V\cap\sigma\not=\emptyset.
$$
We will modify $\pi$ to $\xi$ keeping the last equality because the equality $\xi^{-1}(\tau_{1/2}(X_i))=(V\cap X_i)\times[0,\,1]$ follows from it and hence $\xi$ is a collar on $V$ in $(\overline{\Ima\tau_{1/2}};\Ima\tau_{1/2},\tau_{1/2}(X_i))$ if $\xi$ is a collar on $V$ in $\overline{\Ima\tau_{1/2}}$.\par
 Let $\theta:K''|_{\overline V}\to\{0,1,[0,\,1]\}$ be a simplicial map such that $\theta=0$ on $K^{\prime\prime0}\cap(\overline V-V)$ and $\theta=1$ at the other vertices and set 
$$
\Xi=\{(x,t)\in V\times[0,\,1]:0\le t\le\theta(x)\}.
$$
Note that for $x\in V$, $\theta(x)=0$ if and only if $\pi(\{x\}\times[0,\,1])$ is a point.
\par
 We shrink $V\times[0,\,1]$---the domain of definition of $\pi$---to $\Xi$.
The map of natural modification $\Xi\ni(x,t)\to\pi(x,t/\theta(x))\in\Ima\pi$ is well-defined and extensible to a semialgebraic $C^0$ embedding $\overline\Xi\to\overline{\Ima\pi}$.
However, the map is not necessarily semilinear even if $\overline V=V$ (i.e.\ $\theta\equiv1$).
This is the second problem.
We modify the map as follows.
More generally we can prove the following statement by the Alexander trick.\\
{\it Given a finite cell complex $L$ and a definable $C^0$ embedding $\rho:|L|\to R^n$ such that $\rho(\sigma)$ is polyhedral for each $\sigma\in L$, there exists a PL embedding $\rho':|L|\to R^n$ such that $\rho'(\sigma)=\rho(\sigma)$ for $\sigma\in L$ and $\rho'=\rho$ on $\sigma$ if $\rho|_\sigma$ is PL.}\par
Hence there exists a semilinear homeomorphism $\xi:\Xi\to\Ima\pi$ such that $\Xi(\cdot,0)=\id$ on $V$ and $\xi(\Xi\cap((\sigma\cap V)\times[0,\,1]))=\pi((\sigma\cap V)\times[0,\,1])$ for $\sigma\in K'$ since $\overline{\pi((\sigma\cap V)\times[0,\,1])}$ are polyhedral.\par
 It remains to enlarge $\Xi$ to $V\times [0,\,1]$.
Clearly there exists a semilinear homeomorphism $\xi':V\times[0,\,1]\to\Xi$ of the form $\xi'(x,t)=(\xi''(x,t),t)$ for $(x,t)\in V\times[0,\,1]$ such that 
$$
\xi'((V\cap\sigma)\times[0,\,1])=\Xi\cap((V\cap\sigma)\times[0,\,1])\quad\text{for}\ \sigma\in K'.
$$
Hence $\xi\circ\xi':V\times[0,\,1]\to\overline{\Ima\tau_{1/2}}$ is the required collar on $V$ in $\overline{\Ima\tau_{1/2}}$, and the existence in (i) is proved.\qed\vskip2mm\noindent
{\bf Remark 8.}
{\it The map $\tau^{-1}:\Ima\tau\to X$ in the above proof is extensible to a semialgebraic $C^0$ map $\overline{\tau^{-1}}:\overline{\Ima\tau}\to\overline X$ such that $\overline{\tau^{-1}}(\sigma\cap\overline{\Ima\tau})=\sigma$ for $\sigma\in K'$ with $\Int\sigma\subset X$.}\vskip2mm
 We prepare for proving (ii).
Let $(X;X_i)$ be a finite family of semilinear sets bounded in $R^n$ such that $X$ is not compact, and $K$ be as in the above proof.
Set $W=\overline X-X$ and $U_K=\{x\in|\st(v,K'')|:v\in K^{\prime\prime0}\cap W\}$, i.e.\ the closure of $X-\Ima\tau$ in the above proof, (note that $\Ima\tau=(X-U_K)\cup\{x\in\sigma:\sigma\in K'',\,\sigma\subset\overline W-W\}$), let $L$ be another simplicial decomposition of $\overline X$ having the same property as $K$, let $L_1$ and $L_2$ be derived subdivisions of $L$ and $L_1$, respectively, (we call $L_2$ a {\it double derived subdivision} of $L$) and define $U_L$ by $L_2$ likewise $U_K$ by $K''$.
Then the $(U_K;U_K\cap X,U_K\cap X_i)$ and $(U_L;U_L\cap X,U_L\cap X_i)$ are PL homeomorphic for the following reason.
This is a small generalization of (9.4) of Sect.\ 2 (Theorem 3.24 of \cite{R-S}).
As the proof in the present case is the same, we briefly repeat the proof in \cite{R-S}.
We can assume that $L$ is a subdivision of $K$ by replacing $L$ with a simplicial subdivision of the cell complex $\{\sigma_K\cap\sigma_L:\sigma_K\in K,\,\sigma_L\in L\}$.
Let $\tilde K_1$ and $\tilde K_2$ be any derived subdivisions of $K$ and $\tilde K_1$, respectively, such that $\tilde K_1|_{\overline W}=K'|_{\overline W}$ and $\tilde K_2|_{\overline W}=K''|_{\overline W}$, let $\tilde L_1$ and $\tilde L_2$ be any derived subdivisions of $L$ and $\tilde L_1$, respectively, such that $\tilde L_1|_{\overline W}=L_1|_{\overline W}$ and $\tilde L_2|_{\overline W}=L_2|_{\overline W}$, and define $\tilde U_K$ and $\tilde U_L$ by $\tilde K_2$ and $\tilde L_2$ likewise $U_K$ by $K''$.
Then by (9.1), $(U_K;U_K\cap X,U_K\cap X_i)$ and $(U_L;U_L\cap X,U_L\cap X_i)$ are PL homeomorphic to $(\tilde U_K;\tilde U_K\cap X,\tilde U_K\cap X_i)$ and $(\tilde U_L;\tilde U_L\cap X,\tilde U_L\cap X_i)$, respectively, and by (9.2) there exist $\tilde K_1,\,\tilde L_1,\,\tilde K_2$ and $\tilde L_2$ such that 
$$
(\tilde U_K;\tilde U_K\cap X,\tilde U_K\cap X_i)=(\tilde U_L;\tilde U_L\cap X,\tilde U_L\cap X_i).
$$
Hence $(U_K;U_K\cap X,U_K\cap X_i)$ and $(U_L;U_L\cap X,U_L\cap X_i)$ are PL homeomorphic.
Note that the homeomorphism is extensible to a PL homeomorphism of $(\overline X;X,X_i)$.\par
 We call this property {\it the property of invariance}.
Using this we will show the following remark, by which we can assume that any standard semilinear family is of the form $(\Ima\tau;\tau(X_i))$ when we prove (ii) of Theorem 4.\vskip2mm\noindent
{\bf Remark 9.}
{\it Let $(X;X_i),\,K,\,W,\,U_K$ and $\tau$ be the same as above.
Assume that $(X;X_i)$ is standard.
Then $(X;X_i)$ is semilinearly homeomorphic to $(\Ima\tau;\tau(X_i))$.}\vskip2mm\noindent
{\it Proof of Remark 9.}
Let $\phi:W\times[0,\,1]\to\overline X$ be a collar on $W$ in $(\overline X;X,X_i)$ and let $\overline\phi:\overline W\times[0,\,1]\to\overline X$ be the PL extension of $\phi$ (Remark 3 (i)).
We will reduce the problem to the one on $W\times[0,\,1]$ through $\phi$.
Let $Y$ denote the boundary of $\Ima\phi$ in $\overline X$.
Then $\overline\phi^{-1}(Y)\cap(W\times\{0\})=\emptyset$ since $\phi(W\times[0,\,1))$ is a neighborhood of $W$ in $\overline X$.
Let $h$ be a PL function on $\overline W$ such that $h>0$ on $W$, $h<1$ and the set $Z=\{(x,t)\in W\times[0,\,1]:t\le h(x)\}$ does not intersect with $\overline\phi^{-1}(Y)$.
By the property of invariance we can subdivide $K$ and assume that there is a simplicial decomposition $M$ of $\overline W\times[0,\,1]$ such that $\overline\phi:M\to K$ is simplicial and $Z$ is the union of some open simplexes in $M$.
Let us define $U_M$ by $W\times\{0\}$ in $\overline W\times[0,\,1]$ and the double barycentric subdivision $M''$.
Then $\phi(U_M\cap(W\times(0,\,1]))=U_K\cap X$ and $\phi^{-1}(\Ima\tau)=W\times[0,\,1]-U_M$.
Hence it suffices to find a semilinear homeomorphism $\rho$ from $W\times(0,\,1]$ to $W\times[0,\,1]-U_M$ such that $\rho(\phi^{-1}(X_i))\subset\phi^{-1}(X_i)$ and $\rho=\id$ outside of $Z$.
We define $\rho$ as follows.\par
 Note that $U_M$ is included in the set $\{(x,t)\in\overline W\times[0,\,1]:t\le h(x)/2\}$.
Let $\overline\rho$ be the PL homeomorphism from $(\overline W\times\{0\})\cup\graph h$ to $\partial U_M\cup\graph h$ of the form $\overline\rho(x,t)=(x,\overline\rho_1(x,t))$ such that $\overline\rho=\id$ on $\graph h$, where $\partial U_M$ denotes the boundary of $U_M$ in $\overline W\times[0,\,1]$.
By the Alexander trick we can extend $\overline\rho$ to a PL homeomorphism $\overline\rho:\overline W\times[0,\,1]\to\overline W\times[0,\,1]-\Int U_M$ of the same form such that $\overline\rho=\id$ on $\{(x,t):t\ge h(x)\}$.
Then the restriction $\rho=\overline\rho|_{W\times(0,\,1]}$ fulfills the requirements.\qed\vskip2mm
 We replace $U_K$ with the following set according to the proof of (ii).
Let us define a semialgebraic $C^0$ function $f_K$ on $\overline X-\{x\in\sigma:\sigma\in K'',\,\sigma\subset\overline W-W\}$ by 
$$
f_K=\left \{
\begin{array}{l}\!
0\quad\text{on}\ K^{\prime\prime0}\cap W\\
\!1\quad\text{on}\ K^{\prime\prime0}-\overline W,
\end{array}
\right.
\quad
f_K(\sum_{j=1}^{l_3}t_j v_j)=\sum_{j=l_1+1}^{l_2}t_j/\sum_{j=1}^{l_2}t_j 
$$
for $t_1,...,t_{l_3}\in[0,\,1]$ with $\sum_{j=1}^{l_3} t_j=1$ and $\sum_{j=1}^{l_2}t_j>0$ and for a simplex in $K''$ whose vertices $v_1,...,v_{l_1}$ are in $W$, $v_{l_1+1},...,v_{l_2}$ are outside of $\overline W$ and $v_{l_2+1},...,v_{l_3}$ are in $\overline W-W$.
Then $\overline X-\Dom f_K\subset\overline W-W\subset X$ (the notation $\Dom$ denotes the domain of definition), $f_K$ is PL if and only if $\overline W=W$ (i.e.\ $X$ is locally closed in $R^n$), $\overline{\cup_{t\in[0,\,1)}f^{-1}(t)}=U_K$, the map $f_K:\{\sigma\in K'':\sigma\cap(\overline W-W)=\emptyset\}\to\{[0,\,1],\{0\},\{1\}\}$ is simplicial, $(\overline X;U_K,X,X_i)$ is PL homeomorphic to $(\overline X;\overline{f^{-1}_K([0,t])},X,X_i)$ for any $t\in(0,\,1)$ by the above arguments of the property of invariance because there exists a derived subdivision $L$ of $K'$ such that $\overline{f^{-1}_K([0,\,t])}$ is defined by $L$ likewise $U_K$ by $K''$, and hence we can replace $U_K$ with $\overline{f^{-1}_K([0,\,1/2])}$.
\vskip2mm
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where $K$ is generated by a 3-simplex, $W$ is the union of one open 2-simplex, one open 1-simplex and one 0-simplex in $K$, and $v_j$ point out the vertices in $K'$ only.\par
 There is another reason why we introduce $f_K$.
For two definably homeomorphic standard families of semilinear sets we will define two $f_K$'s as above and prove that the families are semilinearly homeomorphic by comparing two $f_K$'s.
This idea is similar to the idea of applying Triangulation theorem of definable $C^0$ functions in the proof of Uniqueness theorem of definable triangulation.\par
The proof in the non-locally closed case is long but the idea is the same as in the locally closed one.
We separate the proof and first prove the locally closed one to show the idea.\vskip2mm\noindent
 {\it Proof of (ii) of Theorem 4 for locally closed semilinear sets.}
Let $\eta:(Y;Y_i)\to(Z;Z_i)$ be a definable homeomorphism between standard families of semilinear sets in $R^n$.
Let $K_Y$ be a simplicial decomposition of $\overline Y$ such that $Y$ and $Y_i$ are the unions of some open simplexes in $K_Y$.
Set $W_Y=\overline Y-Y$, and define a semialgebraic $C^0$ function $f_Y$ by $K_Y$ likewise $f_K$ by $K$.
Let $K_Z,\,W_Z$ and $f_Z$ be given for $(Z;Z_i)$ in the same way.
Assume that $Y$ is non-compact and locally closed.
We want to compare $f_Y$ and $f_Z$.
For this we need the condition that $\eta$ is extensible to a definable (not necessarily injective) $C^0$ map $\overline\eta:\overline Y\to\overline Z$, which is possible for the following reason.\par
 Set $X=\graph\eta$ and $X_i=\graph\eta|_{Y_i}$, and let $p_Y:(X;X_i)\to(Y;Y_i)$ and $p_Z:(X;X_i)\to(Z;Z_i)$ denote the projections.
Then $p_Y$ and $p_Z$ are extensible to definable $C^0$ maps $\overline{p_Y}:\overline X\to\overline Y$ and $\overline{p_Z}:\overline X\to\overline Z$, respectively.
By Triangulation theorem of definable sets we can regard $\overline X$ as the underlying polyhedron of a finite simplicial complex $K$ such that $X$ and $X_i$ are the unions of some open simplexes in $K$.
Let us define $\tau:(X;X_i)\to(X;X_i)$ as in the above proof.
Then the $(\Ima\tau;\tau(X_i))$ is a standard family of semilinear sets, and $\tau^{-1}|_{\Ima\tau}:\Ima\tau\to X$ is extensible to a semialgebraic $C^0$ map from $\overline{\Ima\tau}$ into $\overline X$ (Remark 8).
Hence $p_Y\circ(\tau^{-1}|_{\Ima\tau}):(\Ima\tau;\tau(X_i))\to(Y;Y_i)$ and $p_Z\circ(\tau^{-1}|_{\Ima\tau}):(\Ima\tau;\tau(X_i))\to(Z;Z_i)$ are definable homeomorphisms between standard families of semilinear sets and are extensible to definable $C^0$ maps $\overline{\Ima\tau}\longrightarrow\overline Y$ and $\overline{\Ima\tau}\longrightarrow\overline Z$, respectively.
Thus by replacing $\eta$ with $p_Y\circ(\tau^{-1}|_{\Ima\tau})$ and $p_Z\circ(\tau^{-1}|_{\Ima\tau})$, we can assume that $\eta$ is extensible to a definable $C^0$ map $\overline\eta:\overline Y\to\overline Z$.\par
 Next, we modify $K''_Y$ and $K''_Z$ to certain cell complexes so that $(Y;Y\cap\sigma:\sigma\in K''_Y)$ becomes standard since $(Y;Y\cap\sigma:\sigma\in K''_Y)$ is not standard if $Y$ is of dimension $>1$ locally at $\overline Y-Y$.
We set
$$
\tilde Y=f^{-1}_Y((1/2,\,1]),\ \tilde Y_i=\tilde Y\cap Y_i,\ \tilde K_Y=\{\sigma\cap f^{-1}_Y([1/2,\,1]),\,\sigma\cap f^{-1}(1/2):\sigma\in K''_Y\},\ \tilde f_Y=2f_Y-1, 
$$
and owing to Remark 9, replace $(Y;Y_i),\,K''_Y$ and $f_Y$ with $(\tilde Y;\tilde Y_i),\,\tilde K_Y$ and $\tilde f_Y$, respectively.
Then we can assume that $\tilde K_Y$ is a cellular (not simplicial if $Y$ is of dimension $>1$ locally at $\overline Y-Y$) decomposition of $\overline Y$, $\tilde f_Y:\tilde K_Y\to\{0,1,[0,\,1]\}$ is a cellular map, for each cell $\sigma$ in $\tilde K_Y$ such that $\sigma\cap W_Y\not=\emptyset$ and $\sigma\not\subset W_Y$, $\sigma\cap\tilde f^{-1}_Y([0,\,1/2])$ is linearly isomorphic to $(\sigma\cap W_Y)\times[0,\,1]$ and $\tilde K_Z$ satisfies the same conditions.
Hence $(Y;Y\cap\sigma:\sigma\in\tilde K_Y)$ and $(Z;Z\cap\sigma:\sigma\in\tilde K_Z)$ are standard.\par
 There are then semialgebraic isotopies $\pi_{Y t}:\overline Y\to\overline Y$ and $\pi_{Z t}:\overline Z\to\overline Z$, $0\le t\le1$, of the identity maps preserving $\{\sigma\cap Y:\sigma\in\tilde K_Y\}$ and $\{\sigma\cap Y:\sigma\in\tilde K_Z\}$, respectively, such that $\Ima\pi_{Y t}=\tilde f^{-1}_Y([t/2,\,1])$ and $\Ima\pi_{Z t}=\tilde f^{-1}_Z([t/2,\,1])$.
Hence if $\tilde f_Y=\tilde f_Z\circ\overline\eta$ on $\tilde f^{-1}_Y([0,\,t_0])$ for some $t_0\in(0,\,1]$ (which is possible as shown below) then there is a definable isotopy $\eta_t:(Y;Y_i)\to(Z;Z_i)$, $0\le t\le1$, of $\eta$ through homeomorphisms such that $\eta_1$ is extensible to a definable homeomorphism from $\overline Y$ onto $\overline Z$.
Therefore, (ii) of Theorem 4 follows from Uniqueness theorem of definable triangulation and its supplement in the case where $Y$ and $Z$ are locally closed and $\tilde f_Y=\tilde f_Z\circ\tilde\eta$ on $\tilde f_Y^{-1}([0,\,t_0])$.\par
 It remains to reduce the problem to the case $\tilde f_Y=\tilde f_Z\circ\overline\eta$ on some neighborhood of $W_Y$ in $\overline Y$.
Set $g_Y=\tilde f_Z\circ\overline\eta$, which is a definable $C^0$ function on $\overline Y$ with zero set $W_Y\,(=\tilde f^{-1}_Y(0))$.
Apply Triangulation theorem of definable $C^0$ functions to $g_Y$.
Then there exists a definable isotopy $\xi_t,\ 0\le t\le1$, of $\overline Y$ preserving $\tilde K_Y$ such that the $g_Y\circ\xi_1$ is PL.
Next, apply (13) of Sect.\ 2 to $\tilde f_Y$ and $g_Y\circ\xi_1$.
Then we have a PL isotopy $\delta_t,\ 0\le t\le1$, of $\overline Y$ preserving $\tilde K_Y$ such that $g_Y\circ\xi_1\circ\delta_1=\tilde f_Y$ on some definable neighborhood of $W_Y$ in $\overline Y$.
Hence translating $\overline\eta$ to $\overline\eta\circ\xi_1\circ\delta_1$ through $\overline\eta\circ\xi_t\circ\delta_t,\ 0\le t\le1$, we can assume that $\tilde f_Y=\tilde f_Z\circ\overline\eta$ around $W_Y$.
Thus (ii) is proved for locally closed semilinear sets.\qed\vskip2mm
 When $Y$ is not locally closed, we proceed as follows.
In this case too, we show that the equality $\tilde f_Y=\tilde f_Z\circ\overline\eta$ can hold on a neighborhood of $W_Y$ in $\overline Y$.
However, the neighborhood is not of the form $\tilde f_Y^{-1}([0,\,t_0]),\ t_0\in(0,\,1)$.
In the last step of the following proof we move the neighborhood to some $\tilde f_Y^{-1}([0,\,t_0])$.\par
 We prepare for the proof of (ii) for non-locally closed semilinear sets.
Let $\eta:(Y;Y_i)\to(Z;Z_i),\,K_Y,\,\allowbreak W_Y,\,f_Y,\,K_Z,\,W_Z$ and $f_Z$ be the same as in the locally closed case.
Note that we can replace $(Y;Y_i)$ or $(Z;Z_i)$ with a standard family of semilinear sets through a semilinear homeomorphism and a definable homeomorphism if the definable homeomorphism is isotopic to a semilinear homeomorphism through homeomorphisms.
By such replacement we will reduce the problem to a simple case.
Let us consider the case where $Y$ is not locally closed.
We will subdivide (or modify) $K''_Y$ and $K''_Z$ into $K_{jY}$ and $K_{jZ},\ j=1,2$, respectively, in sequence, define $f_{jY}$ and $f_{jZ}$ by $K_{j Y}$ and $K_{j Z}$, respectively, likewise $f_Y$ by $K''_Y$ and reduce the problem to the case where the following six conditions are satisfied in order.\vskip2mm\noindent
(a) (i) {\it For any $t\in[0,\,1)$ there exists a natural cellular isomorphism $\kappa_{Y t}$ from $K_{1Y}$ to $\{\sigma\cap(f_{1Y}^{-1}(t)\overline),\sigma\cap(f^{-1}_{1Y}([t,\,1])\overline):\sigma\in K_{1Y}\}$ such that the map $\overline Y\times[0,\,1)\ni(y,t)\to\kappa_{Y t}(y)\in\overline Y$ is continuous,} (ii) {\it the restriction of $f_{1Y}$ to $\{\sigma\in K_{1Y}:\sigma\subset\Dom f_{1Y}\}$ is a cellular map into $\{0,1,[0,\,1]\}$,} (iii) {\it each cell in $K_{1Y}$ is uniquely described as of the form $\sigma_1*\sigma_2$ for $\sigma_1,\sigma_2\in K_{1Y}$ such that $\sigma_1\subset\overline Y-\Dom f_{1Y}$ and $\sigma_2\subset\Dom f_{1Y}$, 
$$
f_{1Y}(ty_1+(1-t)y_2)=f_{1Y}(y_2)\quad\text{for}\ (y_1,y_2,t)\in\sigma_1\times\sigma_2\times[0,\,1)\leqno{\rm(iv)}
$$
for such $\sigma_1$ and $\sigma_2$,} (v) {\it the $\sigma_2\cap f^{-1}_{1Y}([0,\,1/2])$ is linearly isomorphic to $(\sigma_2\cap W_Y)\times[0,\,1]$ for each $\sigma_2\in K_{1Y}$ such that $\sigma_2\not\subset W_Y$, $\sigma_2\subset\Dom f_{1Y}$ and $\sigma_2\cap W_Y\not=\emptyset$,} (vi) {\it the $(Y;Y\cap\sigma:\sigma\in K_{1Y})$ is standard, and (vii) the property of invariance holds for $K_{1Y}$ and any double derived subdivision of $K_Y$.
The $K_{1Z}$ satisfies the same conditions, and $\{\Int\sigma\cap Z:\sigma\in K_{1Z}\}$ is compatible with $\{\eta(\sigma\cap Y):\sigma\in K_{1Y}\}$.}\vskip2mm\noindent
(b) {\it The $\eta$ is extended to a definable homeomorphism $\overline\eta:Y\cup(\overline{W_Y}-W_Y\overline)\to Z\cup(\overline{W_Z}-W_Z\overline)$ and $\overline\eta|_{(\overline{W_Y}-W_Y\overline)}$ is a PL homeomorphism onto $(\overline{W_Z}-W_Z\overline)$, where the notation $(\ \overline)$ denote the closure of a set $(\ )$.}\vskip2mm\noindent
(c) {\it The $K_{1Y}$ and $K_{1Z}$ are subdivided into simplicial complexes $K_{2Y}$ and $K_{2Z}$, the $\overline\eta|_{(\overline W_Y-W_Y\overline)}:K_{2Y}|_{(\overline W_Y-W_Y\overline)}\allowbreak\to K_{2Z}|_{(\overline W_Z-W_Z\overline)}$ is simplicial and $\eta(\overline Y-\Dom f_{2Y})=\overline Z-\Dom f_{2Z}$.}\vskip2mm\noindent
(d) {\it For non-negative PL functions $\phi_Y$ on $\overline Y$ and $\phi_Z$ on $\overline Z$ with zero set $\overline Y-\Dom f_{2Y}$ and $\overline Z-\Dom f_{2Z}$, respectively, the functions $\phi_Y f_{2Y}$ and $(\phi_Z f_{2Z})\circ\eta$ on $Y\cap\Dom f_{2Y}$ are extensible to PL and definable $C^0$ functions on $\overline Y$ with zero set $\overline{W_Y}$, say, $\overline{\phi_Y f_{2Y}}$ and $\overline{(\phi_Z f_{2Z})\circ\eta}$, respectively, $\phi_Y=\phi_Z\circ\eta$ on $Y\cap G^{-1}([0,\,\epsilon]\times[0,\,1])\,(=Y\cap\phi^{-1}_Y([0,\,\epsilon]))$ for some $\epsilon\in(0,\,1)$, and there exists a definable neighborhood $N$ of $(0,\,\epsilon]\times\{0\}$ in $(0,\,\epsilon]\times[0,\,1]$ such that the $\overline{(\phi_Z f_{2Z})\circ\eta}|_{G^{-1}(\overline N)}$ is extensible to a non-negative PL function $\tilde F_2$ on $\overline Y$ with zero set $\overline{W_Y}$, where $G=(G_1,G_2)=(\phi_Y,\overline{\phi_Y f_{2Y}})$.}\vskip2mm\noindent
(e) {\it It holds that $\tilde F_2=G_2$ and hence $\overline{\phi_Y f_{2Y}}=\overline{(\phi_Z f_{2Z})\circ\eta}$ on $G^{-1}(\overline N)$.}\vskip2mm\noindent
(f) {\it The $f_{2Z}\circ\eta$ coincides with $f_{2Y}$ on $Y\cap G^{-1}(\hat N)$ for some definable neighborhood $\hat N$ of $[\epsilon,\,1]\times\{0\}$ in $[\epsilon,\,1]\times[0,\,1]$.}\vskip2mm
 Under (a) we will find a definable isotopy $\alpha_t,\ 0\le t\le1$, of $Y$ preserving $\{Y\cap\sigma:\sigma\in K_{1Y}\}$ by induction on the dimension of $Y$ such that $\eta\circ\alpha_1$ is semilinear.
(We do not directly show that $\eta$ is definably isotopic to a semilinear homeomorphism through homeomorphisms because the induction process becomes complicated for the direct construction.) 
We use the induction hypothesis to obtain (b).\vskip2mm\noindent
{\it Proof that (a) can be satisfied.}
Set
$$
\tilde Y=(\overline Y-\Dom f_Y)\cup f^{-1}_Y((1/2,\,1]),\quad\tilde Y_i=\tilde Y\cap Y_i,
$$
$$
\tilde K_Y=\{\sigma\cap(\tilde Y\overline),\,\big(\sigma\cap f^{-1}_Y(1/2)\overline{\big)}:\sigma\in K''_Y\}\quad\text{and}\quad\tilde f_Y=2f_Y-1.
$$
Then $\tilde K_Y$ is a cell complex, $((\tilde Y\overline);\tilde Y,\tilde Y_i)$ and $(\overline Y;Y,Y_i)$ are PL homeomorphic (Remarks 1 (iii) and 7), $\tilde f_Y$ is defined by $\tilde K_Y$ likewise $f_Y$ by $K''_Y$, $\tilde f_Y:\{\sigma\in\tilde K_Y:\sigma\subset\Dom\tilde f_Y\}\to\{0,\,1,[0,\,1]\}$ is cellular, each cell in $\tilde K_Y$ is of the form either $\sigma_1*(\sigma_2\cap f^{-1}_Y(1/2))$ or $\sigma_1*(\sigma_2\cap f^{-1}_Y([1/2,\,1]))$ for some $\sigma_1,\sigma_2\in K''_Y$ such that $\sigma_1\subset\overline Y-\Dom f_Y$ and $\sigma_2\subset\Dom f_Y$, 
$$
\tilde f_Y(t y_1+(1-t)y_2)=2f_Y(t y_1+(1-t)y_2)-1=2f_Y(y_2)-1=\tilde f_Y(y_2)\qquad\qquad\qquad
$$
$$\qquad\qquad\text{for}\ (y_1,y_2,t)\in\sigma_1\times(\sigma_2\cap f_Y^{-1}(1/2))\times[0,\,1)\ (\text{or}\ \sigma_1\times(\sigma_2\cap f_Y^{-1}([1/2,\,1]))\times[0,\,1))
$$ 
for such $\sigma_1$ and $\sigma_2$, the correspondence $\sigma_3\cap f^{-1}_Y(1/2)\to\sigma_3\cap f^{-1}_Y(1/2+t/2)$ induces a cellular isomorphism $\kappa_{Y t}$ from $\tilde K_Y$ to $\{\sigma\cap(\tilde f^{-1}_Y(t)\overline),\sigma\cap(\tilde f^{-1}_Y([t,\,1])\overline):\sigma\in\tilde K_Y\}$ for each $t\in[0,\,1)$ and each 1-simplex $\sigma_3$ in $K''_Y$ with one vertex in $\Dom f_Y$ and the other in $\overline Y-\Dom f_Y$, the map $(\tilde Y\overline)\times[0,\,1)\ni(y,t)\to\tilde\kappa_{Y t}(y)\in(\tilde Y\overline)$ is continuous, $\sigma_2\cap\tilde f_Y^{-1}([0,\,1/2])$ is linearly isomorphic to $(\sigma_2\cap f^{-1}_Y(1/2))\times[0,\,1]$ for each $\sigma_2\in\tilde K_Y$ such that $\sigma_2\not\subset\overline{f_Y^{-1}(1/2)},\ \sigma_2\subset\Dom\tilde f_Y$ and $\sigma_2\cap f_Y^{-1}(1/2)\not=\emptyset$, and $(\tilde Y;\tilde Y\cap\sigma:\sigma\in\tilde K_Y)$ is standard.
(It is not the case unless we replace $U_K$ with $\overline{f^{-1}_Y([0,\,1/2])}$.) 
Hence, we can replace $(Y;Y_i),\,K''_Y$ and $f_Y$ with $(\tilde Y;\tilde Y_i),\,\tilde K_Y$ and $\tilde f_Y$, respectively.
We use new notation $K_{1Y}$ and $f_{1Y}$ for $\tilde K_Y$ and $\tilde f_Y$ since $K_{1Y}$ is now a cellular decomposition of $\overline Y$.
In the same way we can modify $K''_Z$.
Then the conditions in (a) are satisfied except the one that $\{\Int\sigma\cap Z:\sigma\in K_{1Z}\}$ is compatible with $\{\eta(\sigma\cap Y):\sigma\in K_{1Y}\}$.
For the condition we modify the above construction and $\eta$ as follows.\par
 There is a semialgebraic homeomorphism $\lambda_Y:(\tilde Y;\tilde Y\cap\sigma:\sigma\in\tilde K_Y)\to(Y;Y\cap\sigma:\sigma\in K''_Y)$ such that $\lambda_Y|_{(\tilde Y;\tilde Y_i)}:(\tilde Y;\tilde Y_i)\to(Y;Y_i)$ is definably isotopic to a semilinear homeomorphism $(\tilde Y;\tilde Y_i)\to(Y;Y_i)$ but not $(\tilde Y;\tilde Y\cap\sigma:\sigma\in\tilde K_Y)\to(Y;Y\cap\sigma:\sigma\in K''_Y)$.
First we only replace $(Y;Y_i)$, $K''_Y$ and $f_Y$ with $(\tilde Y;\tilde Y_i)$, $K_{1Y}$ and $f_{1Y}$ and keep the notation $(Y;Y_i)$ and $\eta$ for $(\tilde Y;\tilde Y_i)$ and $\eta\circ\lambda_Y$.
Next, by Triangulation theorem of definable sets there is a definable isotopy $\beta_t,\ 0\le t\le1$, of $\overline Z$ preserving $K_Z$ such that $\beta_1(\overline{\eta(Y\cap\sigma)})\ (\sigma\in K_{1Y}$) are polyhedra.
Thirdly, we replace $\eta$ with $\beta_1\circ\eta$.
Then by subdividing $K_Z$ we can assume that $\{\Int\sigma\cap Z:\sigma\in K_Z\}$ is compatible with $\{\eta(\sigma\cap Y):\sigma\in K_{1Y}\}$.
Fourthly, we define $\tilde K_Z$ and $(\tilde Z;\tilde Z_i)$ by this $K_Z$.
Finally, we replace $(Z;Z_i),\ K''_Z,\ f_Z$ and $\eta$ with $(\tilde Z;\tilde Z_i),\ K_{1Z},\ f_{1Z}$ and $\lambda_Z^{-1}\circ\eta$ and keep the notation $(Z;Z_i)$ and $\eta$, where $\lambda_Z$ is defined in the same way as $\lambda_Y$.
Then the remaining condition in (a) is satisfied.\qed\vskip2mm\noindent
{\it Proof that (b) can be satisfied.}
As we have mentioned, we use the induction hypothesis here.
By Remark 3 (ii), $(\overline{W_Y}-W_Y;Y_i\cap\overline W_Y)$ is standard.
It is easy to also see that $(\overline{W_Y}-W_Y;Y_i\cap\overline{W_Y}),\,K_{1Y}|_{(\overline{W_Y}-W_Y\overline)},\,\eta|_{\overline{W_Y}-W_Y},\allowbreak\,(\overline{W_Z}-W_Z;Z_i\cap\overline{W_Z})$ and $K_{1Z}|_{(\overline{W_Z}-W_Z\overline)}$ satisfy the conditions on $(Y;Y_i),\,K_{1Y},\,\eta,\,(Z;Z_i)$ and $K_{1Z}$ in (a).
Hence by the induction hypothesis there exists a definable isotopy $\alpha_{Wt},\ 0\le t\le 1$, of $\overline{W_Y}-W_Y$ preserving $\{\sigma\cap\overline{W_Y}-W_Y:\sigma\in K_{1Y}\}$ such that $\eta\circ\alpha_{W1}$ is semilinear.
Then by Remark 3 (iii), $\eta\circ\alpha_{W1}$ is extended to a PL homeomorphism from $(\overline{W_Y}-W_Y\overline)$ onto $(\overline{W_Z}-W_Z\overline)$ since $\overline{W_Y}-W_Y$ and $\overline{W_Z}-W_Z$ are standard, and by Lemma 5 we can extend $\alpha_{Wt}$ to a definable isotopy $\tilde\alpha_{Wt},\ 0\le t\le 1$, of $Y$ preserving $\{Y\cap\sigma:\sigma\in K_{1Y}\}$.
(To be precise we construct, by induction on $l\in\N$, a definable isotopy $\alpha_{l t},\ 0\le t\le1$, of $Y\cap|K^l_{1Y}|$ preserving $\{Y\cap\sigma:\sigma\in K^l_{1Y}\}$ such that $\alpha_{l t}$ is an extension of $\alpha_{l-1t}$ and $\alpha_{l t}=\alpha_{W t}$ on $|K^l_{1Y}|\cap\overline W_Y-W_Y$.) 
Hence by replacing $\eta$ with $\eta\circ\tilde\alpha_{W1}$ we can assume (b), i.e.\ that the extension $\overline\eta|_{(\overline{W_Y}-W_Y\overline)}$ of $\eta|_{\overline{W_Y}-W_Y}$ is a PL homeomorphism onto $(\overline{W_Z}-W_Z\overline)$.\qed\vskip2mm\noindent
{\it Proof that (c) can be satisfied.}
Let $\hat K_{1Y}$ and $\hat K_{1Z}$ be simplicial subdivisions of $K_{1Y}$ and $K_{1Z}$, respectively, such that $\overline\eta|_{(\overline{W_Y}-W_Y\overline)}:\hat K_{1Y}|_{(\overline{W_Y}-W_Y\overline)}\to\hat K_{1Z}|_{(\overline{W_Z}-W_Z\overline)}$ is an isomorphism, and set $K_{2Y}=\hat K''_{1Y}$ and $K_{2Z}=\hat K''_{1Z}$.
Then $\eta(\overline Y-\Dom f_{2Y})=\overline Z-\Dom f_{2Z}$.
Hence replacing $K_{1Y},\,f_{1Y},\,K_{1Z}$ and $K_{1Z}$ with $K_{2Y},\,f_{2Y},\,K_{2Z}$ and $K_{2Z}$ we assume (c).\qed\vskip2mm
 Now, we define a simplicial map $\phi_Y:K_{2Y}\to\{0,1,[0,\,1]\}$ by 
$$
\phi_Y=
\left \{
\begin{array}{l}\!
0\quad\text{on}\ K^0_{2Y}\cap(\overline{W_Y}-W_Y)\\
\!1\quad\text{on}\ K^0_{2Y}-(\overline{W_Y}-W_Y)
\end{array}
\right.
$$
and a simplicial map $\phi_Z:K_{2Z}\to\{0,1,[0,\,1]\}$ in the same way for simplicity of notation.
Then $\phi_Y^{-1}(0)=\overline Y-\Dom f_{2Y}$, $\phi_Z^{-1}(0)=\overline Z-\Dom f_{2Z}$, and $\phi_Y f_{2Y}$, $\phi_Z f_{2Z}$ and $(\phi_Z f_{2Z})\circ\eta$ are extensible to a simplicial map $\overline{\phi_Y f_{2Y}}:K_{2Y}\to\{0,1,[0,\,1]\}$ with zero set $\overline{W_Y}$, a simplicial map $\overline{\phi_Z f_{2Z}}:K_{2Z}\to\{0,1,[0,\,1]\}$ with zero set $\overline{W_Z}$ and a definable $C^0$ function $\overline{(\phi_Z f_{2Z})\circ\eta}$ on $\overline Y$ with zero set $\overline{W_Y}$, respectively.
Note that $\phi_Z\circ\eta$ is not necessarily extensible to a definable $C^0$ function on $\overline Y$, and
$$
\Ima(\phi_Y,\overline{\phi_Y f_{2Y}})=\Ima(\phi_Z,\overline{\phi_Z f_{2Z}})=\{(u,v)\in[0,\,1]^2:v\le u\}.
$$
Set $F=(F_1,F_2)=(\phi_Z\circ\eta,\overline{(\phi_Z f_{2Z})\circ\eta})$ and $G=(G_1,G_2)=(\phi_Y,\overline{\phi_Y f_{2Y}})$.\vskip2mm\noindent
{\it Proof that (d) can be satisfied.}
The first condition in (d) is satisfied as we have already mentioned.\par
 For the second it suffices to find a definable isotopy $\mu_t,\ 0\le t\le1$, of $Y$ preserving $\{\sigma\cap Y:\sigma\in K_{2Y}\}$ such that $\phi_Y=\phi_Z\circ\eta\circ\mu_1$ on $Y\cap G^{-1}([0,\,\epsilon]\times[0,\,1])$ for some $\epsilon>0$.
First we reduce the problem to the case where $\phi_Z\circ\eta$ is extensible to a definable $C^0$ function on $\overline Y$, say, $\overline{\phi_Z\circ\eta}$.
Set $X=\graph\phi_Z\circ\eta\subset Y\times R$, and let $\phi_1$ and $\phi_2$ be the projection $\overline X\to R$ and the composite of the projection $\overline X\to\overline Y$ and $\phi_Y$, respectively.
Then only for construction of $\mu_t$ we can replace $\phi_Y:\overline Y\to R$ and $\phi_Z\circ\eta:Y\to R$ with $\phi_2$ and $\phi_1$, respectively.
Hence we can assume that $\phi_Z\circ\eta$ is extensible to a definable $C^0$ function on $\overline Y$.
Here $\overline Y$ becomes a compact definable set and $\phi_Y$ becomes a definable $C^0$ function on $\overline Y$.
However, by Triangulation theorem of definable $C^0$ functions we can regard $\phi_Y$ as a PL function on a compact polyhedron $\overline Y$ and $\sigma\cap Y$ for each $\sigma$ in the original $K_{2Y}$ as the union of some open simplexes in a new simplicial complex $K_{2Y}$.
By Triangulation theorem of definable $C^0$ functions there is a definable isotopy $\nu_t,\ 0\le t\le1$, of $\overline Y$ preserving $K_{2Y}$ such that $\overline{\phi_Z\circ\eta}\circ\nu_1$ is PL.
Hence we assume from the beginning that $\overline{\phi_Z\circ\eta}$ is PL.
Then by (13) of Sect.\ 2 we can suppose that $\phi_Y=\phi_Z\circ\eta$ on $Y\cap G^{-1}([0,\,\epsilon]\times[0,\,1])$ for some $\epsilon>0$.\par
 The third condition is satisfied by Lemma 7, and (d) can be satisfied.\qed\vskip2mm\noindent
{\it Proof that (e) can be satisfied.} 
We will find a PL homeomorphism $\delta$ of $\overline Y$ and a PL isotopy whose finishing homeomorphism is $\delta$ such that $\tilde F_2\circ\delta=G_2$ on $G^{-1}(\overline N)$ (i.e.\ $\overline{(\phi_Z f_{2Z})\circ\eta}\circ\delta=\overline{\phi_Y f_{2Y}}$).
Then the equality $\phi_Y\circ\delta=\phi_Y$ should hold on $G^{-1}(\overline N)$.
If we consider $\tilde F_2$ and $G_2$ on $\phi^{-1}_Y(\epsilon')$ only for one $\epsilon'\in(0,\,1]$ and construct a PL homeomorphism $\delta_{\epsilon'}$ of $\phi^{-1}_Y(\epsilon')$ such that $\tilde F_2\circ\delta_{\epsilon'}=G_2$ on $G^{-1}(\overline N)\cap\phi^{-1}_Y(\epsilon')$, then the equality $\phi_Y\circ\delta_{\epsilon'}=\phi_Y$ is obvious.
We will choose $\epsilon'$ so close to 0 that such $\delta_{\epsilon'}$ is extensible to $\overline Y$ and the equality holds on $G^{-1}([0,\,\epsilon']\times[0,\,1])$.\par
 Let $K_{3Y}$ be a simplicial subdivision of $K_{2Y}$ such that $G|_{K_{3Y}}$ is a simplicial map onto some simplicial complex and the restriction of $\tilde F_2$ to each simplex in $K_{3Y}$ is linear (we cannot find $K_{3Y}$ such that both of $\tilde F_2|_{K_{3Y}}$ and $G|_{K_{3Y}}$ are simplicial maps onto some simplicial complexes, and a counterexample is given before Lemma 2.1 of \cite{S6}).
Let $\epsilon'\in R$ be positive and so small that $\phi_Y(K^0_{3Y})\cap(0,\,\epsilon']=\emptyset$, and let $K_{4Y}$ be the canonical simplicial subdivision of the cell complex $\{\sigma\cap\phi^{-1}_Y(\sigma_{\epsilon'}):\sigma\in K_{3Y},\,\sigma_{\epsilon'}\in\{0,\epsilon',1,[0,\,\epsilon'],[\epsilon',\,1]\}\}$.
Set $K_{4Y\epsilon'}=K_{4Y}|_{\phi^{-1}_Y(\epsilon')}$, and compare $\tilde F_2$ and $G_2$ on $\phi^{-1}_Y(\epsilon')$.
(Note that the $\{G(\sigma)\cap([0,\,\epsilon']\times[0,\,1]):\sigma\in K_{4Y}\}$ is a simplicial complex generated by the simplexes $0*G(\sigma),\ \sigma\in K_{4Y\epsilon'}$.) 
Since both are PL and non-negative and have the same zero set there exists a PL homeomorphism $\delta_{\epsilon'}$ (and an isotopy) of $\phi^{-1}_Y(\epsilon')$ preserving $K_{4Y\epsilon'}$ such that $\tilde F_2\circ\delta_{\epsilon'}=G_2$ on a neighborhood of $\phi^{-1}_Y(\epsilon')\cap\overline{W_Y}$ in $\phi^{-1}_Y(\epsilon')$, say, $G^{-1}(O)$ for a small segment $O$ with one end $(\epsilon',0)$ and the other in $\{\epsilon'\}\times[0,\,1]$.
We need to extend $\delta_{\epsilon'}$ to a PL homeomorphism $\delta$ of $\overline Y$ preserving $K_{4Y}$.
It is obvious on $\phi^{-1}_Y([\epsilon',\,1])$ by the Alexander trick.
We define $\delta$ on $\phi^{-1}_Y([0,\,\epsilon'])$ by 
$$
\delta(ry_1+(1-r)y_2)=ry_1+(1-r)\delta_{\epsilon'}(y_2)\qquad\qquad\qquad\qquad\qquad
$$
$$
\qquad\text{for}\ (y_1,y_2,r)\in\sigma_1\times\sigma_2\times[0,\,1],\ \sigma_1\in K_{4Y}|_{\phi^{-1}_Y(0)},\ \sigma_2\in K_{4Y\epsilon'}
\ \text{with}\ \sigma_1*\sigma_2\in K_{4Y}.
$$
(Any simplex in $K_{4Y}|_{\phi_Y^{-1}([0,\,\epsilon'])}$ is of form $\sigma_1*\sigma_2$ for such $\sigma_1$ and $\sigma_2$.) 
Then for such $(y_1,y_2,r)$ with $G(y_2)\in O$, 
$$
\tilde F_2\circ\delta(r y_1+(1-r)y_2)=\tilde F_2(ry_1+(1-r)\delta_{\epsilon'}(y_2))=r\tilde F_2(y_1)+(1-r)\tilde F_2\circ\delta_{\epsilon'}(y_2)=\qquad\qquad
$$
$$
\qquad\qquad\qquad\qquad\qquad(1-r)\tilde F_2\circ\delta_{\epsilon'}(y_2)=(1-r)G_2(y_2)=G_2(r y_1+(1-r)y_2), 
$$
since $\tilde F_2$ and $G_2$ are linear on $\sigma_1*\sigma_2$.
Hence $\tilde F_2\circ\delta=G_2$ on $G^{-1}((0,0)*O)$.
Here the cone $(0,0)*O$ is a semilinear neighborhood of $(0,\,\epsilon']\times\{0\}$ in $[0,\,\epsilon']\times[0,\,1]$.
On the other hand, $\phi_Y\circ\delta(r y_1+(1-r)y_2)=(1-r)\epsilon'=\phi_Y(r y_1+(1-r)y_2)$ on $G^{-1}([0,\,\epsilon]\times[0,\,1])$.
Therefore, by shrinking $N$ to $((0,0)*O)\cap N$ and keeping the same notation $\epsilon$ and $N$ we can assume (e), i.e.\ that $\tilde F_2=G_2$ on $G^{-1}((0,0)*O)$ and $F=G$ on $G^{-1}(\overline N)$.\qed\vskip2mm\noindent
{\it Proof that (f) can be satisfied.} 
Note that $f_{2Z}\circ\eta|_{Y\cap G^{-1}([\epsilon,\,1]\times[0,\,1])}$ is extended to a definable $C^0$ function on $G^{-1}([\epsilon,\,1]\times[0,\,1])$.
Let $g_Z$ denote the extension and set $g_Y=f_{2Y}|_{G^{-1}([\epsilon,\,1]\times[0,\,1])}$.
Then $g_Z=g_Y$ on $G^{-1}(N\cap(\{\epsilon\}\times[0,\,1]))$, and it suffices to find a definable isotopy $\pi_t,\,0\le t\le1$, of $G^{-1}([\epsilon,\,1]\times[0,\,1])$ preserving $K_{4Y}|_{G^{-1}([\epsilon,\,1]\times[0,\,1])}$ such that $g_Z\circ\pi_1=g_Y$ on $G^{-1}([\epsilon,\,1]\times[0,\,\epsilon''])$ for some $\epsilon''>0$ because $\pi_t,\,0\le t\le1$, is extensible to a definable isotopy $\tilde\pi_t,\,0\le t\le1$, of $\overline Y$ preserving $K_{4Y}$ such that $\phi_Y\circ\tilde\pi_1=\phi_Y$ on $G^{-1}([0,\,\epsilon]\times[0,\,1])$ and $\overline{(\phi_Z f_{2Z})\circ\eta}\circ\tilde\pi_1=\overline{\phi_Y f_{2Y}}$ on $G^{-1}(\overline N)$ by the same reason as in the above proof.
Here only for construction of $\pi_Y$ we can assume that $g_Y$ and $g_Z$ are PL by Triangulation theorem of definable $C^0$ functions (1) and (2) and by the same reason as above.
Then the existence of $\pi_t$ follows from (13) of Sect.\ 2 because $g_Y$ and $g_Z$ satisfy the conditions in (13).
Hence (f) can be satisfied.\qed\vskip2mm\noindent
{\it Proof of (ii) in Theorem 4 for non-closed semilinear sets.}
As we have already shown, we can assume (a),...,(f).
We will find a definable isotopy $\alpha_t,\ 0\le t\le 1$, of $Y$ preserving $\{Y\cap\sigma:\sigma\in K_{1Y}\}$ such that $\eta\circ\alpha_1:(Y;Y_i)\to(Z;Z_i)$ is semilinear.\par
 First we see that $(\overline Y;Y,Y_i)$ and $(\overline Z;Z,Z_i)$ are definably homeomorphic.
Set $G_Z=(\phi_Z,\overline{\phi_Z f_{2Z}})$ (the definable $C^0$ extension of $(\phi_Z,\phi_Z f_{2Z})$ to $\overline Z$), and let $L$ be the simplicial complex generated by the simplex $\{(u,v)\in[0,\,1]^2:v\le u\}$.
Let us remember that $G:K_{2Y}\to L$ and $G_Z:K_{2Z}\to L$ are simplicial.
Let $\psi_1$ and $\psi_2$ be definable non-negative $C^0$ functions defined on $[0,\,1]$ with zero set $\{0\}$ such that $\psi_1(u)<u$ and $\psi_2(u)<u$ for $u\in(0,\,1]$, 
and set 
$$
\Psi_j=\{(u,v)\in[0,\,1]^2:\psi_j(u)\le v\le u\},\quad j=1,2.
$$
Note that $G^{-1}(\Psi_j)\subset Y$ and $G^{-1}_Z(\Psi_j)\subset Z$.
Later we define $\psi_1$ and $\psi_2$ more explicitly so that the $G^{-1}(\Psi_1)$ and $G^{-1}_Z(\Psi_1)$ are definably homeomorphic to $\overline Y$ and $\overline Z$, respectively, and $\eta|_{G(\Psi_2)}$ is a homeomorphism onto $G_Z^{-1}(\Psi_2)$.
Hence we need the following lemma, which we will prove later.\vskip2mm\noindent
{\bf Lemma 10.}
There exist definable isotopies $\delta_{Y t},\ 0\le t\le 1$, of $\overline Y$ preserving $K_{2Y}$ and $\delta_{Z t},\ 0\le t\le 1$, of $\overline Z$ preserving $K_{2Z}$ such that $\delta_{Y1}(G^{-1}(\Psi_1))=G^{-1}(\Psi_2)$, $\delta_{Z1}(G_Z^{-1}(\Psi_1))=G_Z^{-1}(\Psi_2)$, $\delta_{Y t}=\id$ on a definable neighborhood of $W_Y$ in $\overline Y$ and $\delta_{Z t}=\id$ on a definable neighborhood of $W_Z$ in $\overline Z$.\vskip2mm
{\it Continued proof of (ii) in Theorem 4 for non-closed semilinear sets.}
Assume that Lemma 10 is proved.
Choose $\psi_1$ and $\psi_2$ so that $\psi_1(u)=d u$ for some small positive $d\in R$, $\psi_2=\psi_1$ on $[\epsilon,\,1]$ and $\{(u,v)\in(0,\,1]^2:v\le\psi_2(u)\}$ is included in $N\cup(\hat N\cap G_Z\circ\eta(Y\cap G^{-1}(\hat N)))$.
Then $G^{-1}(\Psi_1)=\overline{f_{2Y}([d,\,1])}$ and $G^{-1}_Z(\Psi_1)=\overline{f_{2Z}([d,\,1])}$ by definition of $G$ and $G_Z$; hence there are PL homeomorphisms $\kappa_Y:\overline Y\to G^{-1}(\Psi_1)$ and $\kappa_Z:\overline Z\to G^{-1}_Z(\Psi_1)$ by (a); moreover, $\eta(G^{-1}(\Psi_2))=G^{-1}_Z(\Psi_2)$ for the following reason.\par
 Since the boundaries of $G^{-1}(\Psi_2)$ in $\overline Y$ and $G^{-1}_Z(\Psi_2)$ in $\overline Z$ are $\{y\in Y:\phi_Y f_{2Y}(y)=\psi_2\circ\phi_Y(y)\}$ and $\{z\in Z:\phi_Zf_{2Z}(z)=\psi_2\circ\phi_Z(z)\}$, respectively, it suffices to see 
$$
\{y\in Y:\phi_Y f_{2Y}(y)=\psi_2\circ\phi_Y(y)\}=\{y\in Y:(\phi_Z f_{2Z})\circ\eta(y)=\psi_
2\circ\phi_Z\circ\eta(y)\}.\leqno{\text{(g)}}
$$
We consider (g) on $\phi^{-1}_Y(0),\ \phi^{-1}_Y((0,\,\epsilon])$ and $\phi^{-1}_Y([\epsilon,\,1])$ separately.
First the following is obvious 
$$
\{y\in\phi^{-1}_Y(0):\phi_Y f_{2Y}(y)=\psi_2\circ\phi_Y(y)\}=\phi^{-1}_Y(0)=\{y\in\phi^{-1}_Y(0):(\phi_Z f_{2Z})\circ\eta(y)=\psi_2\circ\phi_Z\circ\eta(y)\}.
$$
Next, we prove
$$
\{y\in\phi^{-1}_Y((0,\,\epsilon]):\phi_Y f_{2Y}(y)=\psi_2\circ\phi_Y(y)\}=\{y\in\phi^{-1}_Y((0,\,\epsilon]):(\phi_Z f_{2Z})\circ\eta(y)=\psi_2\circ\phi_Z\circ\eta(y)\}.
$$
Let $y$ be an element of the left-side set.
Then 
$$
G(y)=(\phi_Y(y),\phi_Y f_{2Y}(y))=(\phi_Y(y),\psi_2\circ\phi_Y(y))\in N.
$$
$$
(\phi_Y(y),\phi_Y f_{2Y}(y))=(\phi_Z\circ\eta(y),(\phi_Z f_{2Z})\circ\eta(y))\quad\text{by (d) and (e).}\leqno{\text{Hence}}
$$\vskip-7mm
$$
(\phi_Z f_{2Z})\circ\eta(y)=\phi_Y f_{2Y}(y)=\psi_2\circ\phi_Y(y)=\psi_2\circ\phi_Z\circ\eta(y).\leqno{\text{Therefore}}
$$
Namely, $y$ is an element of the right-side set.
In the same way we see that the right-side set is included in the left-side.\par
 Finally, we see (g) on $\phi^{-1}_Y([\epsilon,\,1])$.
In the same way as above, by using (f) and the inclusions $\{(u,v)\in[\epsilon,\,1]\times(0,\,1]:v\le\psi_2(u)\})\subset\hat N$ and $G^{-1}_Z(\{(u,v)\in[\epsilon,\,1]\times(0,\,1]:v\le\psi_2(u)\})\subset\eta(Y\cap G^{-1}(\hat N))$, we obtain 
$$
\{y\in\phi^{-1}_Y([\epsilon,\,1]):f_{2Y}(y)=d\}=\{y\in\phi^{-1}_Y([\epsilon,\,1]):f_{2Z}\circ\eta(y)=d\}.
$$\vskip-7mm
$$
\{y\in\phi^{-1}_Y([\epsilon,\,1]):\phi_Y f_{2Y}(y)=\psi_2\circ\phi_Y(y)\}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leqno{\text{Hence}}
$$
$$
=\{y\in\phi^{-1}_Y([\epsilon,\,1]):f_{2Y}(y)=d\}\quad\text{since}\ \psi_2(u)=d u\ \text{for}\ u\in[\epsilon,\,1]\quad
$$
$$
=\{y\in\phi^{-1}_Y([\epsilon,\,1]):f_{2Z}\circ\eta(y)=d\}
$$
$$
\qquad\qquad=\{y\in\phi^{-1}_Y([\epsilon,\,1]):(\phi_Z f_{2Z})\circ\eta(y)=\psi_2\circ\phi_Z\circ\eta(y)\}\quad\text{since}\ \psi_2(u)=d u\ \text{for}\ u\in[\epsilon,\,1].
$$
Therefore, (g) holds.\par
 Set $\omega_Y=\delta_{Y1}$ and $\omega_Z=\delta_{Z1}$.
Then we have a sequence of definable homeomorphisms
$$
\overline Y\stackrel{\kappa_Y}{\longrightarrow}G^{-1}(\Psi_1)\stackrel{\omega_Y}{\longrightarrow}G^{-1}(\Psi_2)\stackrel{\eta}{\longrightarrow}G^{-1}_Z(\Psi_2)\stackrel{\omega_Z^{-1}}{\longrightarrow}G^{-1}_Z(\Psi_1)\stackrel{\kappa_Z^{-1}}{\longrightarrow}\overline Z.
$$
Moreover, the composite $\kappa^{-1}_Z\circ\cdots\circ\kappa_Y$ carries $\sigma\cap Y$ onto $\eta(\sigma\cap Y)$ for each $\sigma\in K_{1Y}$ by the condition in (a) that $\{\Int\sigma\cap Z:\sigma\in K_{1Z}\}$ is compatible with $\{\eta(\sigma\cap Y):\sigma\in K_{1Y}\}$.
Hence the composite is a definable homeomorphism from $(\overline Y;Y,Y_i)$ to $(\overline Z;Z,Z_i)$.\par
 Next, we see that the restriction to $Y$ of the composite is the finishing homeomorphism of some isotopy of $\eta:Y\to Z$ through homeomorphisms.
For this we find isotopies whose finishing homeomorphisms are $\kappa_Y,\ \omega_Y,\ \kappa_Z$ and $\omega_Z$, and construct the required isotopy $\alpha_t$ of $Y$ by the isotopies.
Let us define $\psi_{j t}$ and $\Psi_{j t}$ to be $t\psi_j$ and $\{(u,v)\in[0,\,1]^2:\psi_{j t}(u)\le v\le u\}$, respectively, for each $t\in[0,\,1]$ and $j=1,2$, and repeat the above arguments for $\psi_{j t}$ and $\Psi_{j t}$ for $t\in(0,\,1]$.
Then by the above proof we have definable isotopies $\omega_{Y t},\ 0\le t\le 1$, of $\overline Y$ preserving $K_{2Y}$ and $\omega_{Z t},\ 0\le t\le 1$, of $\overline Z$ preserving $K_{2Z}$ such that $\omega_{Y t}(G^{-1}(\Psi_{1t}))=G^{-1}(\Psi_{2t})$ and $\omega_{Z t}(G^{-1}_Z(\Psi_{1t}))=G^{-1}_Z(\Psi_{2t})$ for $t\in[0,\,1]$.
On the other hand, by (a) there are definable isotopies $\kappa_{Y t}$ and $\kappa_{Z t},\ 0\le t\le1$, of the identity maps of $\overline Y$ and $\overline Z$ preserving $\{Y\cap\sigma:\sigma\in K_{1Y}\}$ and $\{Z\cap\sigma:\sigma\in K_{1Z}\}$, respectively, such that $\Ima\kappa_{Y t}=\overline{f_{2Y}([t d,\,1])}$ and $\Ima\kappa_{Z t}=\overline{f_{2Z}([t d,\,1])}$.
Set $\kappa'_{Y t}=\omega_{Y t}\circ\kappa_{Y t}$ and $\kappa'_{Z t}=\omega_{Z t}\circ\kappa_{Z t}$.
Then $\kappa'_{Y t}$ and $\kappa'_{Z t}$, $0\le t\le 1$, are definable isotopies of the identity maps of $\overline Y$ and $\overline Z$ preserving $\{Y\cap\sigma:\sigma\in K_{1Y}\}$ and $\{Z\cap\sigma:\sigma\in K_{1Z}\}$, respectively, such that $\Ima\kappa'_{Y t}=G^{-1}(\Psi_{2t})$ and $\Ima\kappa'_{Z t}=G^{-1}_Z(\Psi_{2t})$.
Note that $\Ima\eta\circ\kappa'_{Y t}=\Ima\kappa'_{Z t}$ for each $t\in(0,\,1]$.
Hence $\kappa^{\prime-1}_{Z t}\circ\eta\circ\kappa'_{Y t}|_Y:(Y;Y_i)\to(Z;Z_i)$, $0\le t\le1$, is a definable isotopy of $\eta$ to $\kappa_{Z1}^{\prime-1}\circ\eta\circ\kappa'_{Y1}|_Y$ through homeomorphisms and $\{\Int\sigma\cap Z:\sigma\in K_{1Z}\}$ is compatible with $\{\kappa^{\prime-1}_{Z t}\circ\eta\circ\kappa'_{Y t}(\sigma\cap Y):\sigma\in K_{1Y}\}$ by (a) for each $t$.\par
 If $\kappa^{\prime-1}_{Z1}\circ\eta\circ\kappa'_{Y1}$ is PL, then we define $\alpha_t$ to be $\eta^{-1}\kappa_{Z t}^{\prime-1}\circ\eta\circ\kappa'_{Y t}|_Y$.
Then $\alpha_t$ fulfills the requirements that $\alpha_t,\ 0\le t\le 1$, is a definable isotopy of $Y$ preserving $\{Y\cap\sigma:\sigma\in K_{1Y}\}$ and $\eta\circ\alpha_1$ is semi-linear.
If $\kappa^{\prime-1}_{Z1}\circ\eta\circ\kappa'_{Y1}$ is not PL, then we modify it to a PL homeomorphism.
By Uniqueness theorem of definable triangulation and its supplement, there is a definable isotopy $\mu_t:(\overline Y;Y\cap\sigma:\sigma\in K_{1Y})\to(\overline Z;\eta(Y\cap\sigma):\sigma\in K_{1Y}),\ 0\le t\le1$, of $\kappa^{\prime-1}_{Z1}\circ\eta\circ\kappa'_{Y1}$ through homeomorphisms such that $\mu_1$ is PL since $\kappa_{Z1}^{\prime-1}\circ\eta\circ\kappa'_{Y1}(Y\cap\sigma)=\eta(Y\cap\sigma)$ for $\sigma\in K_{1Y}$ and $\eta(Y\cap\sigma)$ are semilinear by (a).
We define $\alpha_t$ by
$$
\alpha_t=\left \{
\begin{array}{l}
\eta^{-1}\circ\kappa^{\prime-1}_{Z2t}\circ\eta\circ\kappa'_{Y2t}|_Y\hspace{7mm}\text{for}\ t\in[0,\,1/2]\\
\eta^{-1}\circ\mu_{2t-1}|_Y\hspace{21mm}\text{for}\ t\in[1/2,\,1].
\end{array}
\right.
$$
Then $\alpha_t$ fulfills the requirements by the same reason as above.
Thus (ii) in Theorem 4 for non-locally closed semilinear sets is proved.\qed\vskip2mm\noindent
{\it Proof of Lemma 10.} 
We construct $\delta_{Yt}$ only.
Let $\psi_0$ and $\psi_3$ be functions having the same properties as $\psi_1$ and $\psi_2$ such that $\psi_0<\psi_j<\psi_3,\ j=1,2$, on $(0,\,1]$.
First define a definable isotopy $\delta_{L t},\ 0\le t\le 1$, of $|L|$ preserving $L$ so that for each $(u,v)\in|L|$, 
$$
\delta_{L t}(u,v)=
\left \{
\begin{array}{l}
\!(u,v)\hspace{34mm}\text{for}\ v\in[0,\,\psi_0(u)]\cup[\psi_3(u),\,u]\\
\!\big(u,t\psi_2(u)+(1-t)\psi_1(u)\big)\hspace{3mm}\text{for}\ v=\psi_1(u),
\end{array}
\right.
$$
and $\delta_{L t}(u,\cdot)$ is linear on the segments $[\psi_0(u),\,\psi_1(u)]$ and $[\psi_1(u),\,\psi_3(u)]$.
Then $\delta_{L1}(\Psi_1)=\Psi_2$.
Next, lift $\delta_{L t}$ to a definable isotopy $\delta_{Yt}$ of $\overline Y$ preserving $K_{2Y}$ by Lemma 6.
Then $\delta_{Y1}(G^{-1}(\Psi_1))=G^{-1}(\Psi_2)$ and $\delta_{Y t}=\id$ on $G^{-1}(\{v\le\psi_0(u)\})$---a definable neighborhood of $W_Y$ in $\overline Y$.
Thus Lemma 10 is proved.\qed
{\small\begin{thebibliography}{99}
%\small{\bibitem{B}E.~Baro, \textit{Normal triangulations in o-minimal structures}, J.~Symbolic Logic, 75(2010), 275-286.
%\bibitem{B-C-R}J.~Bochnak, M.~Coste and M.F.~Roy, \textit{Real algebraic geometry}, Springer Erg.~d.~Math., 36, Springer-Verlag, 1998.
%\bibitem{B-O1}A.~Berarducci and M.~Otero, \textit{O-minimal fundamental group, homology and manifolds}, J.~London Math.~Soc., 65(2002), 257-270.
%\bibitem{B-O2}A.~Berarducci and M.~Otero, \textit{Transfer methods for o-minimal topology}, J.~Symbolic Logic, 68(2003), 785-794.
%\bibitem{B-O3}A.~Berarducci and M.~Otero, \textit{Topology of definable abelian groups in o-minimal structure}, Bull.~London Math.~Soc., 44(2012), 473-479.
%\bibitem{C1}M.~Coste, \textit{Unicit\' e des triangulations semi-alg\' ebriques\,:\ validit\' e sur un corps r\' eel clos quelconque, et effectivit\' e forte}, C.~R.~Acad.~Sci.~Paris, S\' er.~I Math., 312(1991), 395-398.
%\bibitem{C2}M.~Coste, \textit{An introduction to o-minimal geometry}, Dottorato di Ricerca in.~Matematica, Dip.~Mat.~Univ.~Pisa.~Instituti Editoriali e Poligrafici Internazion, 2000.
%\bibitem{C-S1}M.~Coste and M.~Shiota, \textit{Nash triviality in families of Nash manifolds}, Invent.~Math., 108(1992), 349-368.
%\bibitem{C-S2}M.~Coste and M.~Shiota, \textit{Thom's first isotopy lemma\,:\ a semialgebraic version, with uniform bound Real analytic and algebraic geometry}, Real Analytic and Algebraic Geometry, Walter de Gruyter, 83-101, 1995.
%\bibitem{H}J.~F.~P.~Hudson, \textit{Piecewise linear topology}, Benjamin, 1969.
\bibitem{K-S1}R.C.~Kirby and L.C.~Siebenmann, \textit{On the triangulation of manifolds and the Hauptvermutung}, Bull.~Amer.~Math.~Soc., 75(1969), 742-749.
%\bibitem{K-S2}R.C.~Kirby and L.C.~Siebenmann, \textit{Foundational essays on topological manifolds, smoothings, and triangulations}, Princeton University Press, 1977.
\bibitem{K-P-S}J.~F.~Knight, A.~Pillay and C.~Steinhorn, \textit{Definable Sets in Ordered Structures II}, Trans.~Amer.~Math.~Soc., 295(1986), 593-605.
%\bibitem{Loi}T.L.~Loi, \textit{Verdier and strict Thom stratifications in o-minimal structures}, Illinois J.~Math., 42(1998), 347-356.
%\bibitem{L1}S.~\L ojasiewicz, \textit{Triangulation of semianalytic sets}, Ann.\ Scuola Norm.\ Sup.\ Pisa, Sci.\ Fis.\ Mat., 18(1964), 449-474.
%\bibitem{L2} S.~\L ojasiewicz, \textit{Ensembles semi-analytiques}, IHES, 1965.
\bibitem{Ma}B.~Mazur, \textit{A note on some contractible 4-manifolds}, Ann.~Math., 73(1961), 221-228.
%\bibitem{M}J.W.~Milnor, \textit{Two complexes which are homeomorphic but combinatorially distinct}, Ann.~of Math., 75(1961), 575-590.
%\bibitem{Mu}J.R.~Munkres, {Elementary differential topology}, Princeton University Press, 1966.
\bibitem{P-S}A.~Pillay and C.~Steinhorn, \textit{Definable Sets in Ordered Structures I}, Trans.~Amer.~Math.~Soc., 295(1986), 565-592.
%\bibitem{R}A.A.~Ranicki (editor), \textit{The Hauptvermutung book: a collection of papers of the topology of manifolds}, Kluwer Academic Publishers, 1996.
\bibitem{R-S-W}J.P.~Rolin, P.~Speissegger and A.J.~Wilkie, \textit{Quasianalytic Denjoy-Carleman class and o-minimality}, J.~Amer.~Math.~Soc., 16(2003), 751-777.
\bibitem{R-S}C.~P.~Rourke and B.~J.~Sanderson, \textit{Introduction to piecewise-linear topology}, Springer, 1972.
%\bibitem{S1}M.~Shiota, \textit{Piecewise linearization of real analytic functions}, Publ.~RIMS, Kyoto Univ., 20(1984), 727-792.
%\bibitem{S2}M.~Shiota, \textit{Nash manifolds}, Lecture Notes in Math., 1269, Springer, 1987.
%\bibitem{S3}M.~Shiota, \textit{Piecewise linearization of real-valued subanalytic functions}, Trans.~Amer.~Math.~Soc., 312(1989), 663-679.
%\bibitem{S4}M.~Shiota, \textit{Piecewise linearization of subanalytic functions II}, Real analytic and algebraic geometry, Trento, 1988, Springer Lecture Notes in Math., 1240, 1990, 247-307.
\bibitem{S5}M.~Shiota, \textit{Geometry of subanalytic and semialgebraic sets}, Birk\-h\"auser Progress in Math., 150, 1997.
\bibitem{S6}M.~Shiota, \textit{O-minimal Hauptvermutung for polyhedra I}, Invent. math., DOI 10.1007/s00222-013-0467-y.
%\bibitem{S-Y}M.~Shiota and M.~Yokoi, \textit{Triangulations of subanalytic sets and locally subanalytic manifold}, Trans.~Amer.~Math.~Soc., 286(1984), 727-750.
%\bibitem{T}B.~Teissier, \textit{Sur la triangulation des morphismes sous-analytiques}, Inst.\ Hautes \' Etudes Sci.\ Publ.\ Math., 70(1989), 169-198.
%\bibitem{v1}L.~van den Dries, \textit{Remarks on Tarski's problem concerning (R, +, $\cdot$, exp)}, Logic Colloquium 1982, Elsevier Science Publishers B.V.~(North-Holland), 1984, 97-121.
%\bibitem{v2}L.~van den Dries, \textit{Tarski problem and Pfaffian functions}, Logic Colloquium 1984, Elsevier Science Publishers B.V.~(North-Holland), 1986, 59-90.
\bibitem{v3}L.~van den Dries, \textit{Tame topology and o-minimal structures}, Cambridge University Press, London Mathematical Society Lecture Note, 248, 1998.
%\bibitem{W}A.J.~Wilkie, \textit{Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function}, J.~Amer.~Math.~Soc., 9(1996), 1051-109.}
\end{thebibliography}
\qquad Graduate school of mathematics, Nagoya University, Chikusa, Nagoya, 464-8602, Japan\\
\qquad{\it E-mail address}: shiota@math.nagoya-u.ac.jp}
\enddocument
1310.5489.extracted_bib
\begin{thebibliography}{99}
%\small{\bibitem{B}E.~Baro, \textit{Normal triangulations in o-minimal structures}, J.~Symbolic Logic, 75(2010), 275-286.
%\bibitem{B-C-R}J.~Bochnak, M.~Coste and M.F.~Roy, \textit{Real algebraic geometry}, Springer Erg.~d.~Math., 36, Springer-Verlag, 1998.
%\bibitem{B-O1}A.~Berarducci and M.~Otero, \textit{O-minimal fundamental group, homology and manifolds}, J.~London Math.~Soc., 65(2002), 257-270.
%\bibitem{B-O2}A.~Berarducci and M.~Otero, \textit{Transfer methods for o-minimal topology}, J.~Symbolic Logic, 68(2003), 785-794.
%\bibitem{B-O3}A.~Berarducci and M.~Otero, \textit{Topology of definable abelian groups in o-minimal structure}, Bull.~London Math.~Soc., 44(2012), 473-479.
%\bibitem{C1}M.~Coste, \textit{Unicit\' e des triangulations semi-alg\' ebriques\,:\ validit\' e sur un corps r\' eel clos quelconque, et effectivit\' e forte}, C.~R.~Acad.~Sci.~Paris, S\' er.~I Math., 312(1991), 395-398.
%\bibitem{C2}M.~Coste, \textit{An introduction to o-minimal geometry}, Dottorato di Ricerca in.~Matematica, Dip.~Mat.~Univ.~Pisa.~Instituti Editoriali e Poligrafici Internazion, 2000.
%\bibitem{C-S1}M.~Coste and M.~Shiota, \textit{Nash triviality in families of Nash manifolds}, Invent.~Math., 108(1992), 349-368.
%\bibitem{C-S2}M.~Coste and M.~Shiota, \textit{Thom's first isotopy lemma\,:\ a semialgebraic version, with uniform bound Real analytic and algebraic geometry}, Real Analytic and Algebraic Geometry, Walter de Gruyter, 83-101, 1995.
%\bibitem{H}J.~F.~P.~Hudson, \textit{Piecewise linear topology}, Benjamin, 1969.
\bibitem{K-S1}R.C.~Kirby and L.C.~Siebenmann, \textit{On the triangulation of manifolds and the Hauptvermutung}, Bull.~Amer.~Math.~Soc., 75(1969), 742-749.
%\bibitem{K-S2}R.C.~Kirby and L.C.~Siebenmann, \textit{Foundational essays on topological manifolds, smoothings, and triangulations}, Princeton University Press, 1977.
\bibitem{K-P-S}J.~F.~Knight, A.~Pillay and C.~Steinhorn, \textit{Definable Sets in Ordered Structures II}, Trans.~Amer.~Math.~Soc., 295(1986), 593-605.
%\bibitem{Loi}T.L.~Loi, \textit{Verdier and strict Thom stratifications in o-minimal structures}, Illinois J.~Math., 42(1998), 347-356.
%\bibitem{L1}S.~\L ojasiewicz, \textit{Triangulation of semianalytic sets}, Ann.\ Scuola Norm.\ Sup.\ Pisa, Sci.\ Fis.\ Mat., 18(1964), 449-474.
%\bibitem{L2} S.~\L ojasiewicz, \textit{Ensembles semi-analytiques}, IHES, 1965.
\bibitem{Ma}B.~Mazur, \textit{A note on some contractible 4-manifolds}, Ann.~Math., 73(1961), 221-228.
%\bibitem{M}J.W.~Milnor, \textit{Two complexes which are homeomorphic but combinatorially distinct}, Ann.~of Math., 75(1961), 575-590.
%\bibitem{Mu}J.R.~Munkres, {Elementary differential topology}, Princeton University Press, 1966.
\bibitem{P-S}A.~Pillay and C.~Steinhorn, \textit{Definable Sets in Ordered Structures I}, Trans.~Amer.~Math.~Soc., 295(1986), 565-592.
%\bibitem{R}A.A.~Ranicki (editor), \textit{The Hauptvermutung book: a collection of papers of the topology of manifolds}, Kluwer Academic Publishers, 1996.
\bibitem{R-S-W}J.P.~Rolin, P.~Speissegger and A.J.~Wilkie, \textit{Quasianalytic Denjoy-Carleman class and o-minimality}, J.~Amer.~Math.~Soc., 16(2003), 751-777.
\bibitem{R-S}C.~P.~Rourke and B.~J.~Sanderson, \textit{Introduction to piecewise-linear topology}, Springer, 1972.
%\bibitem{S1}M.~Shiota, \textit{Piecewise linearization of real analytic functions}, Publ.~RIMS, Kyoto Univ., 20(1984), 727-792.
%\bibitem{S2}M.~Shiota, \textit{Nash manifolds}, Lecture Notes in Math., 1269, Springer, 1987.
%\bibitem{S3}M.~Shiota, \textit{Piecewise linearization of real-valued subanalytic functions}, Trans.~Amer.~Math.~Soc., 312(1989), 663-679.
%\bibitem{S4}M.~Shiota, \textit{Piecewise linearization of subanalytic functions II}, Real analytic and algebraic geometry, Trento, 1988, Springer Lecture Notes in Math., 1240, 1990, 247-307.
\bibitem{S5}M.~Shiota, \textit{Geometry of subanalytic and semialgebraic sets}, Birk\-h\"auser Progress in Math., 150, 1997.
\bibitem{S6}M.~Shiota, \textit{O-minimal Hauptvermutung for polyhedra I}, Invent. math., DOI 10.1007/s00222-013-0467-y.
%\bibitem{S-Y}M.~Shiota and M.~Yokoi, \textit{Triangulations of subanalytic sets and locally subanalytic manifold}, Trans.~Amer.~Math.~Soc., 286(1984), 727-750.
%\bibitem{T}B.~Teissier, \textit{Sur la triangulation des morphismes sous-analytiques}, Inst.\ Hautes \' Etudes Sci.\ Publ.\ Math., 70(1989), 169-198.
%\bibitem{v1}L.~van den Dries, \textit{Remarks on Tarski's problem concerning (R, +, $\cdot$, exp)}, Logic Colloquium 1982, Elsevier Science Publishers B.V.~(North-Holland), 1984, 97-121.
%\bibitem{v2}L.~van den Dries, \textit{Tarski problem and Pfaffian functions}, Logic Colloquium 1984, Elsevier Science Publishers B.V.~(North-Holland), 1986, 59-90.
\bibitem{v3}L.~van den Dries, \textit{Tame topology and o-minimal structures}, Cambridge University Press, London Mathematical Society Lecture Note, 248, 1998.
%\bibitem{W}A.J.~Wilkie, \textit{Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function}, J.~Amer.~Math.~Soc., 9(1996), 1051-109.}
\end{thebibliography}