On $k_\mathbb{R}$-spaces and $s_\mathbb{R}$-spaces

TL;DR

This paper characterizes $k_ eal$-spaces and $s_ eal$-spaces, providing conditions under which the space of continuous functions $C_p(X)$ exhibits these properties, including special cases and consistency results under the continuum hypothesis.

Contribution

It offers new characterizations of $k_ eal$-spaces and $s_ eal$-spaces and identifies conditions for $C_p(X)$ to have these properties, including examples under set-theoretic assumptions.

Findings

$C_p(X)$ is a $k_ eal$-space for spaces with one non-isolated point

If $|X|$ is not sequential, then $C_p(X)$ is an $s_ eal$-space

Under $(CH)$, there exists a space where $C_p(X)$ is Ascoli but not $k_ eal$

Abstract

We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or an $s_\mathbb{R}$-space. It is proved that $C_p(X)$ is a $k_\mathbb{R}$-space for any space $X$ with one non-isolated point; if, in addition, $|X|$ is not sequential, then $C_p(X)$ is even an $s_\mathbb{R}$-space. Under $(CH)$, it is shown that there exists a separable metrizable space $X$ such that $C_p(X)$ is an Ascoli space but not a $k_\mathbb{R}$-space.