  Given a subset of $X\subseteq \mathbb{R}^{n}$ we can associate with every
point $x\in \mathbb{R}^{n}$ a vector space $V$ of maximal dimension with the
property that for some ball centered at $x$, the subset $X$ coincides inside
the ball with a union of lines parallel with $V$. A point is singular if $V$
has dimension $0$. In an earlier paper we proved that a $(\mathbb{R}, +,<
,\mathbb{Z})$-definable relation $X$ is actually definable in $(\mathbb{R}, +,<
,1)$ if and only if the number of singular points is finite and every rational
section of $X$ is $(\mathbb{R}, +,< ,1)$-definable, where a rational section is
a set obtained from $X$ by fixing some component to a rational value. Here we
show that we can dispense with the hypothesis of $X$ being $(\mathbb{R}, +,<
,\mathbb{Z})$-definable by assuming that the components of the singular points
are rational numbers. This provides a topological characterization of
first-order definability in the structure $(\mathbb{R}, +,< ,1)$. It also
allows us to deliver a self-definable criterion (in Muchnik's terminology) of
$(\mathbb{R}, +,< ,1)$- and $(\mathbb{R}, +,< ,\mathbb{Z})$-definability for a
wide class of relations, which turns into an effective criterion provided that
the corresponding theory is decidable. In particular these results apply to the
class of $k-$recognizable relations on reals, and allow us to prove that it is
decidable whether a $k-$recognizable relation (of any arity) is
$l-$recognizable for every base $l \geq 2$.
