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

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 $ε>0, \{x\in X: \lvert f(x)\rvert\geq ε\}$ is a member of $\mathcal{P}$. Let $\upsilon_{C_{\mathcal{P}}}X$ stand for the set of all those points $p$ in $βX$ 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 $βX$ 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 $C(X)$.