A number of properties enjoyed by two specially constructed topologies on $C(X)$

TL;DR

This paper investigates properties of generalized topologies on the space of continuous functions, establishing conditions under which these topologies are first countable, second countable, hemicompact, or coincide, based on properties of the underlying space.

Contribution

It introduces and analyzes the properties of $m^I$ and $u^I$ topologies on $C(X)$, generalizing classical topologies, and characterizes when these topologies are countable, hemicompact, or coincide.

Findings

The $m^I$-topology is first countable iff it equals the $u^I$-topology iff $X$ is $I$-pseudocompact.

The $u$-topology and $m$-topology coincide iff $X$ is pseudocompact.

The $m^I$-topology is second countable iff $X$ is compact, metrizable, and $I=C(X)$.

Abstract

If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$, the $m^I$-topology and $u^I$-topology on $C(X)$, generalizing the well-known $m$-topology and $u$-topology in $C(X)$ respectively are already there in the literature. It is proved amongst others that the $m^I$-topology is first countable if and only if the $u^I$-topology= $m^I$-topology on $C(X)$ if and only if $X$ is $I$-$pseudocompact$. A special case of this result on choosing $I=C(X)$ reads: the $u$-topology and $m$-topology on $C(X)$ coincide if and only if $X$ is pseudocompact. It is established that the $m^I$-topology on $C(X)$ is second countable if and only if it is $\aleph_0$-$bounded$ if and only if $X$ is compact, metrizable and $I=C(X)$. Furthermore it is realized that the $m^I$ topology on $C(X)$ is hemicompact if and only if it is $\sigma$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$.