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First-countability, ω-Rudin spaces and well-filtered determined spaces
Citation
Xu, X., Shen, C., Xi, X., & Zhao, D. (2021). First-countability, ω-Rudin spaces and well-filtered determined spaces. Topology and its Applications, 300, Article 107775. https://doi.org/10.1016/j.topol.2021.107775
Abstract
In this paper, we investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties. It is proved that every space with a first-countable sobrification is an ω-Rudin space and every first-countable space is well-filtered determined. Therefore, every ω-well-filtered space with a first-countable sobrification is sober. It is also shown that every irreducible closed subset in a first-countable ω-well-filtered space is countably directed, hence every first-countable ω*-well-filtered d-space is sober.
Publisher
Elsevier
Journal
Topology and its Applications