ag.algebraic geometry - Real spectrum of ring of continuous semialgebraic functions

I don't agree with the preceding answer.

When $U$ is a locally compact semialgebraic set, then $widetilde{U}$ equipped with its sheaf of semi-algebraic continuous functions is isomorphic to the affine scheme $mathrm{Spec}(S^0(U))$. This is proposition 6 in Carral, Coste : Normal spectral spaces and their dimensions, J. Pure Appl. Algebra 30 (1983) 227-235. In particular $widetilde{U}$ is homeomorphic to the prime spectrum of $S^0(U)$, which is homeomorphic to its real spectrum. In case $U$ is not locally compact, the situation is different; there are more points in $mathrm{Spec}(S^0(U))$.