  In this work we prove that a semialgebraic set $M\subset{\mathbb R}^m$ is
determined (up to a semialgebraic homeomorphism) by its ring ${\mathcal S}(M)$
of (continuous) semialgebraic functions while its ring ${\mathcal S}^*(M)$ of
(continuous) bounded semialgebraic functions only determines $M$ besides a
distinguished finite subset $\eta(M)\subset M$. In addition it holds that the
rings ${\mathcal S}(M)$ and ${\mathcal S}^*(M)$ are isomorphic if and only if
$M$ is compact. On the other hand, their respective maximal spectra $\beta_s M$
and $\beta_s^* M$ endowed with the Zariski topology are always homeomorphic and
topologically classify a `large piece' of $M$. The proof of this fact requires
a careful analysis of the points of the remainder $\partial M:=\beta_s^*
M\setminus M$ associated with formal paths.
