Characterizing P-spaces X in Terms of C_p(X)

J. C. Ferrando ; J. Kakol ; S. A. Saxon

Journal of convex analysis, Tome 22 (2015) no. 4, pp. 905-915

Résumé

Dual weak barrelledness led us to prove that $X$ is a $P$-space if and only if every pointwise eventually zero sequence in $C_{p}(X)$ is summable, and other better known characterizations. Novel ones recall utility functions from economics and Arkhangel'skii's (strict) $\tau$-continuity. Mackey $\aleph_0$-barrelled duality leads us to prove that $X$ is discrete if and only if every bounded $\sigma$-compact set in $C_{p}(X)$ is relatively compact. We relax the $\sigma$-compact hypothesis of Velichko and the $\sigma$-countably compact hypothesis of Tkachuk/Shakhmatov to prove\,: {\it X is a P-space if and only if $C_{p}(X)$ is $\sigma$-relatively sequentially complete}.

Détail

Classification : 54C35, 46A08
Mots-clés : P-spaces, relatively compact, weak barrelledness