A Note On Separating Function Sets

TL;DR

This paper investigates conditions under which spaces of continuous functions on a topological space have point-separating subspaces with desirable metric properties, focusing on the relationship between $C_p(X)$ and $C_p^2(X)$ for zero-dimensional spaces.

Contribution

It establishes necessary and sufficient conditions for the existence of metric, point-separating subspaces in $C_p(X)$ and $C_p^2(X)$, especially for zero-dimensional spaces.

Findings

$C_p(X)$ has a discrete point-separating space iff $C_p^2(X)$ does for zero-dimensional $X$

Provides criteria for $C_p(X)$ and $C_p^2(X)$ to have metric point-separating subspaces

Identifies conditions linking properties of $C_p(X)$ and $C_p^2(X)$

Abstract

We study separating function sets. We find some necessary and sufficient conditions for $C_p(X)$ or $C_p^2(X)$ to have a point-separating subspace that is a metric space with certain nice properties. One of the corollaries to our discussion is that for a zero-dimensional $X$, $C_p(X)$ has a discrete point-separating space if and only if $C_p^2(X)$ does.