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

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.