On $a$-Kasch spaces

Estaji, Ali Akbar ; Henriksen, Melvin

Archivum mathematicum, Tome 46 (2010) no. 4, pp. 251-262.

Résumé

If $X$ is a Tychonoff space, $C(X)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C(X)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\bar{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $\operatorname{gen}\,(I)$ then $I$ is a non-essential ideal. We show that $X$ is an $\aleph _0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an $\aleph _1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F(X)$ denote the socle of $C(X)$. For a topological space $X$ with only a finite number of isolated points, we show that $X$ is an $a$-Kasch space if and only if $\frac{C(X)}{C_F(X)}$ is an $a$-Kasch ring.

Détail

Classification : 13A30, 16S60, 46J10, 54C40
Keywords: $a$-Kasch space; almost $P$-space; basically disconnected; $C$-embedded; essential ideal; extremally disconnected; fixed ideal; free ideal; Kasch ring; $P$-space; pseudocompact space; Stone-Čech compactification; socle; realcompactification