A Generalization of the m-Topology on C(X) Finer than the m-Topology

Abstract

It is well known that the component of the zero function in $C(X)$ with the $m$-topology is the ideal $C_\psi(X)$. Given any ideal $I\subseteq C_\psi(X)$, we are going to define a topology on $C(X)$ namely the $m^I$-topology, finer than the $m$-topology in which the component of $0$ is exactly the ideal $I$ and $C(X)$ with this topology becomes a topological ring. We show that compact sets in $C(X)$ with the $m^I$-topology have empty interior if and only if $X\setminus\bigcap Z[I]$ is infinite. It is also shown that nonzero ideals are never compact, the ideal $I$ may be locally compact in $C(X)$ with the $m^I$-topology and every Lindel\"{o}f ideal in this space is contained in $C_\psi(X)$. Finally, we give some relations between topological properties of the spaces $X$ and $C_m(X)$. For instance, we show that the set of units is dense in $C_m(X)$ if and only if $X$ is strongly zero-dimentiona and we characterize the space $X$ for which the set $r(X)$ of regular elements of $C(X)$ is dense in $C_m(X)$.