October 20, 2025Open Access

On kR-spaces and sR-spaces

Key Points

C_p(X) is shown to be a k_space for any space X with one non-isolated point, enhancing our understanding of topological properties.
Under the additional condition that |X| is not sequential, C_p(X) is even proven to be an s_space, expanding its applicability in topology.
The research reveals that, under the axiom of choice (CH), a separable metrizable space can make C_p(X) an Ascoli space but not a k_space.
These new characterizations of k_space and s_space provide sufficient and necessary conditions that affect the behavior of C_p spaces.

Abstract

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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