On strong R-spaces

2026/07/22 by Xinpeng Wen, Meng Bao, Xiaoquan Xu · 1 voice
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Abstract

In this paper, we mainly investigate some basic properties of strong R-spaces. It is shown that the property of being a strong R-space is closed-hereditary, saturated-hereditary and retractive, but not finite productive. Hence the category S-Top_r of strong R-spaces and continuous mappings is not reflective in the category Top₀ of T₀-spaces and continuous mappings. It is proved that a T₀-space (X, τ) is a strong R-space iff every nonempty τ-closed subset of X is compact in (X, τ^d), where τ^d is the de Groot dual of τ; consequently, if (X, τ) is a strong R-space (especially, if (X, τ) is a coherent well-filtered space), then τ⊆ τ^dd. Therefore, for any locally compact strong R-space (X, τ), we have τ=τ^dd. Finally, we investigate conditions under which the Smyth power space and Scott power space of a T₀-space is a strong R-space. Several such conditions are given.

Discussions

• Xinpeng Wen, Meng Bao, Xiaoquan Xu: On strong $R$-spaces https://arxiv.org/abs/2607.20294 https://arxiv.org/pdf/2607.20294 https://arxiv.org/html/2607.20294 [bsky, 0 points, 0 comments]

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