  Let $X$ be a topological space. A subset of $C(X)$, the space of continuous
real-valued functions on $X$, is a partially ordered set in the pointwise
order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$
are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem
of characterizing the order isomorphisms (order preserving bijections) between
$A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and
when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order
isomorphism between $A(X)$ and $A(Y)$ gives rise to an associated homeomorphism
between $X$ and $Y$. This generalizes a classical result of Kaplansky
concerning linear order isomorphisms between $C(X)$ and $C(Y)$ for compact
Hausdorff $X$ and $Y$. The class of near vector lattices is introduced in order
to extend the result further to noncompact spaces $X$ and $Y$. The main
applications lie in the case when $X$ and $Y$ are metric spaces. Looking at
spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz
functions, spaces of differentiable functions, and the bounded, "local" and
"bounded local" versions of these spaces, characterizations of when spaces of
one type can be order isomorphic to spaces of another type are obtained.
