A generalized topological Rudin lemma and sober spaces

Xiaoquan Xu; Yi Yang; Xintong Yang; Xiaoquan Xu; Yi Yang; Xintong Yang

doi:10.3934/math.2026638

AIMS Mathematics

2026, Volume 11, Issue 6: 15513-15523. doi: 10.3934/math.2026638

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Research article

A generalized topological Rudin lemma and sober spaces

1.
School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330000, China
2.
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China

Received: 31 March 2026 Revised: 18 May 2026 Accepted: 21 May 2026 Published: 02 June 2026
MSC : 54D99, 54B20, 06B35, 06F30

In this paper, we introduce and study the notion of $ F $-families in $ T_0 $-spaces. This concept unifies several important classes of objects in domain theory and non-Hausdorff topology, including irreducible sets, filtered families of upper sets, Scott open filters, and irreducible subsets in Smyth power spaces. Using $ F $-families, we establish a generalized topological Rudin lemma that extends the classical topological Rudin lemma to a broader setting. As an application, we obtain some characterizations of sober spaces, from which the Hofmann-Mislove theorem and Heckman-Keimel-Schalk theorem can be directly deduced.
- generalized topological Rudin lemma,
- $ F $-family,
- Scott open filter,
- irreducible set,
- sober space
Citation: Xiaoquan Xu, Yi Yang, Xintong Yang. A generalized topological Rudin lemma and sober spaces[J]. AIMS Mathematics, 2026, 11(6): 15513-15523. doi: 10.3934/math.2026638

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Abstract

In this paper, we introduce and study the notion of $ F $-families in $ T_0 $-spaces. This concept unifies several important classes of objects in domain theory and non-Hausdorff topology, including irreducible sets, filtered families of upper sets, Scott open filters, and irreducible subsets in Smyth power spaces. Using $ F $-families, we establish a generalized topological Rudin lemma that extends the classical topological Rudin lemma to a broader setting. As an application, we obtain some characterizations of sober spaces, from which the Hofmann-Mislove theorem and Heckman-Keimel-Schalk theorem can be directly deduced.

References

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