More on generalizations of topology of uniform convergence and $m$-topology on $C(X)$

TL;DR

This paper explores various generalizations of topologies on the space of continuous functions, providing classifications, introducing new topologies, and analyzing their properties and implications for the structure of $C(X)$.

Contribution

It introduces the $U^I$-topology, characterizes when it coincides with the $m^I$-topology, and studies the denseness of units and zero divisors in different topologies on $C(X)$.

Findings

Necessary and sufficient conditions for $U^I$-topology to match $m^I$-topology.

Denseness of units in $C_U(X)$ linked to strong zero dimensionality of $X$.

Closure descriptions of $C_ ext{P}(X)$ in various topologies.

Abstract

This paper conglomerates our findings on the space $C(X)$ of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the $m$-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of $Z$-ideals of $C(X)$ induced by the $U_I$ and the $m_I$-topologies on $C(X)$. Motivated by the definition of $m^I$-topology, another generalization of the topology of uniform convergence, called $U^I$-topology, is introduced here. Among several other results, it is established that for a convex ideal $I$, a necessary and sufficient condition for $U^I$-topology to coincide with $m^I$-topology is the boundedness of $X\setminus\bigcap Z[I]$ in $X$. As opposed to the case of the $U_I$-topologies (and $m_I$-topologies), it is proved that each $U^I$-topology (respectively, $m^I$-topology) on $C(X)$ is uniquely determined by the ideal $I$. In the last section, the denseness of the set of units of $C(X)$ in $C_U(X)$ (= $C(X)$ with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space $X$. Also, the space $X$ is a weakly P-space if and only if the set of zero divisors (including 0) in $C(X)$ is closed in $C_U(X)$. Computing the closure of $C_\mathscr{P}(X)$ (=$\{f\in C(X):\text{the support of }f\in\mathscr{P}\}$ where $\mathscr{P}$ denotes the ideal of closed sets in $X$) in $C_U(X)$ and $C_m(X)$ (= $C(X)$ with the $m$-topology), the results $cl_UC_\mathscr{P}(X) = C_\infty^\mathscr{P}(X)$ ($=\{f\in C(X):\forall n\in\mathbb{N}, \{x\in X:|f(x)|\geq\frac{1}{n}\}\in\mathscr{P}\}$) and $cl_mC_\mathscr{P}(X)=\{f\in C(X):f.g\in C^\mathscr{P}_\infty(X)\text{ for each }g\in C(X)\}$ are achieved.