On kR-spaces and sR-spaces

Abstract

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