Some new results on functions in $C(X)$ having their support on ideals of closed sets

TL;DR

This paper investigates functions in $C(X)$ supported on ideals of closed sets, characterizes certain realcompact spaces, and explores algebraic properties of related function ideals, extending classical results in topology and ring theory.

Contribution

It provides new characterizations of realcompact spaces via support ideals and analyzes algebraic structures of function ideals in $C(X)$, including conditions for freeness and essentiality.

Findings

Realcompact spaces between $X$ and $eta X$ are characterized by pseudocompactness.

Conditions for product spaces to preserve local-$ extbf{P}$ properties are established.

Criteria for $C_ extbf{P}(X)$ to be a free or essential ideal are identified.

Abstract

For any ideal $\mathcal{P}$ of closed sets in $X$, let $C_\mathcal{P}(X)$ be the family of those functions in $C(X)$ whose support lie on $\mathcal{P}$. Further let $C^\mathcal{P}_\infty(X)$ contain precisely those functions $f$ in $C(X)$ for which for each $\epsilon >0, \{x\in X: \lvert f(x)\rvert\geq \epsilon\}$ is a member of $\mathcal{P}$. Let $\upsilon_{C_{\mathcal{P}}}X$ stand for the set of all those points $p$ in $\beta X$ at which the stone extension $f^*$ for each $f$ in $C_\mathcal{P}(X)$ is real valued. We show that each realcompact space lying between $X$ and $\beta X$ is of the form $\upsilon_{C_\mathcal{P}}X$ if and only if $X$ is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-$\mathcal{P}/$ almost locally-$\mathcal{P}$, becomes a space of the same form. We further show that $C_\mathcal{P}(X)$ is a free ideal ( essential ideal ) of $C(X)$ if and only if $C^\mathcal{P}_\infty(X)$ is a free ideal ( respectively essential ideal ) of $C^*(X)+C^\mathcal{P}_\infty(X)$ when and only when $X$ is locally-$\mathcal{P}$ ( almost locally-$\mathcal{P}$). We address the problem, when does $C_\mathcal{P}(X)/C^\mathcal{P}_{\infty}(X)$ become identical to the socle of the ring $C(X)$. Finally we observe that the ideals of the form $C_\mathcal{P}(X)$ of $C(X)$ are no other than the $z^\circ$-ideals of $C(X)$.