A Note on Minimal Separating Function Sets

TL;DR

This paper investigates minimal point-separating function sets in compact spaces, establishing their equivalence with minimal separating collections of open sets and exploring conditions influencing their existence.

Contribution

It characterizes when minimal separating function sets exist in $C_p(X)$ for compact spaces and links this to properties of open set collections and the structure of $X^2$.

Findings

Equivalence between minimal separating function sets and minimal separating open set collections in compact spaces.

Identification of a visual property of $X^2$ related to the existence of minimal separating functions.

Discussion of open questions and future directions in the study of minimal separating function sets.

Abstract

We study point-separating function sets that are minimal with respect to the property of being separating. We first show that for a compact space $X$ having a minimal separating function set in $C_p(X)$ is equivalent to having a minimal separating collection of functionally open sets in $X$. We also identify a nice visual property of $X^2$ that may be responsible for the existence of a minimal separating function family for $X$ in $C_p(X)$. We then discuss various questions and directions around the topic.