This paper presents new characterizations of k-space and s-space, revealing conditions for C_p spaces.
We give new characterizations of spaces X which are kR-spaces or sR-spaces. Applying the obtained results we provide some sufficient and necessary conditions on X for which Cₚ(X) is a kR-space or an sR-space. It is proved that Cₚ(X) is a kR-space for any space X with one non-isolated point; if, in addition, $|X|$ is not sequential, then Cₚ(X) is even an sR-space. Under $(CH)$, it is shown that there exists a separable metrizable space X such that Cₚ(X) is an Ascoli space but not a kR-space.
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Gabriyelyan et al. (2025) studied this question.
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