In 1988 it was proved by the first author that the closure of a partially semialgebraic set is partially semialgebraic. The essential tool used in that proof was the regular separation property. Here we give another proof without using this tool, based on the semianalytic L-cone theorem (Theorem 2), a semianalytic analog of the Cartan-Remmert-Stein lemma with parameters.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-4-4,
author = {Mar\'\i a-Angeles Zurro},
title = {Closure Theorem for Partially Semialgebraic Sets},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {55},
year = {2007},
pages = {325-331},
zbl = {1158.32003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-4-4}
}
María-Angeles Zurro. Closure Theorem for Partially Semialgebraic Sets. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 325-331. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-4-4/