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

TL;DR

This work introduces and analyzes two generalized topologies, $u^I$ and $m^I$, on $C(X)$ determined by an ideal $I\subseteq C(X)$, and explores how these function-space topologies reflect the underlying topology of $X$. By constructing a concrete metric that induces the $u^I$-topology and establishing extensive equivalences, the authors connect metrizability, first countability, and the coincidence of $u^I$ and $m^I$ to the notion of $I$-pseudocompactness; special cases recover classical results when $I=C(X)$. The paper then derives precise cardinal function equalities for $C_{m^I}(X)$, analyzes boundedness and compactness in this space, and investigates connectedness and completeness properties, including Čech-completeness and hereditary Baireness, under various assumptions on $I$. Overall, the results reveal a rich interplay between topological properties of $X$, the algebraic structure of $C(X)$ relative to $I$, and the induced topologies on the corresponding function spaces, offering a broad framework for understanding when $u^I$ and $m^I$ coincide and when they exhibit favorable metrizability, boundedness, and completeness characteristics.

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 $σ$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$.

Figures (1)

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Theorems & Definitions (65)

• Theorem 3.1
• proof
• Remark 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Theorem 3.6