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Proposition 1.13 of these [notes][1] by Coste implies (if I've read it correctly) that any semialgebraic set $S$ is homeomorphic to a union $U$ of open simplices in some finite simplicial complex $K$.

Thus $S$ is an open subset of an ENR, hence is an ENR. Its singular cohomology therefore coincides with its Čech cohomology, which is finitely generated. Therefore its singular homology must be finitely generated.

Edit: The argument given above is incomplete, since a union of "open" simplices in a simplicial complex is not necessarily open (an easy mistake to make!). However it is easy to see that $S\subseteq\mathbb{R}^n$ is an ENR by other means (in particular, it is locally compact and locally contractible).
[1]: http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf

Proposition 1.13 of these notes by Coste implies (if I've read it correctly) that any semialgebraic set $S$ is homeomorphic to a union $U$ of open simplices in some finite simplicial complex $K$.

Thus $S$ is an open subset of an ENR, hence is an ENR. Its singular cohomology therefore coincides with its Čech cohomology, which is finitely generated. Therefore its singular homology must be finitely generated.

Edit: The argument given above is incomplete, since a union of "open" simplices in a simplicial complex is not necessarily open (an easy mistake to make!). However it is easy to see that $S\subseteq\mathbb{R}^n$ is an ENR by other means (in particular, it is locally compact and locally contractible).

Proposition 1.13 of these [notes][1] by Coste implies (if I've read it correctly) that any semialgebraic set $S$ is homeomorphic to a union $U$ of open simplices in some finite simplicial complex $K$.

Thus $S$ is an open subset of an ENR, hence is an ENR. Its singular cohomology therefore coincides with its Čech cohomology, which is finitely generated. Therefore its singular homology must be finitely generated. [1]: http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf

Proposition 1.13 of these [notes][1] by Coste implies (if I've read it correctly) that any semialgebraic set $S$ is homeomorphic to a union $U$ of open simplices in some finite simplicial complex $K$.

Thus $S$ is an open subset of an ENR, hence is an ENR. Its singular cohomology therefore coincides with its Čech cohomology, which is finitely generated. Therefore its singular homology must be finitely generated.

Edit: The argument given above is incomplete, since a union of "open" simplices in a simplicial complex is not necessarily open (an easy mistake to make!). However it is easy to see that $S\subseteq\mathbb{R}^n$ is an ENR by other means (in particular, it is locally compact and locally contractible).
[1]: http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf

Proposition 1.13 of these [notes][1] by Coste implies (if I've read it correctly) that any semialgebraic set $S$ is homeomorphic to a union $U$ of open simplices in some finite simplicial complex $K$.

Thus $S$ is an open subset of an ANR, hence is an ANR. Its singular cohomology therefore coincides with its Cech cohomology, which is finitely generated. Therefore its singular homology must be finitely generated. [1]: http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf

Proposition 1.13 of these [notes][1] by Coste implies (if I've read it correctly) that any semialgebraic set $S$ is homeomorphic to a union $U$ of open simplices in some finite simplicial complex $K$.

Thus $S$ is an open subset of an ENR, hence is an ENR. Its singular cohomology therefore coincides with its Čech cohomology, which is finitely generated. Therefore its singular homology must be finitely generated. [1]: http://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf