On $a$-Kasch spaces

Ali Akbar Estaji and Melvin Henriksen

Department of Mathematics, Sabzevar Tarbiat Moallem University, P. O. Box 397, Sabzevar, Iran
Harvey Mudd College, Claremont, CA 91711, USA


Abstract: 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 $\operatorname{gen}\,(I)<a$ 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.

AMSclassification: primary 54C40; secondary 13A30.

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.