On the embedding of the Ω-saturation of a topological space

  • Aliaksandr S. Biadrytski Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus https://orcid.org/0000-0001-6187-2871
  • Vladimir L. Timokhovich Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
DOI: https://doi.org/10.33581/2520-6508-2022-1-21-25

Abstract

The countably-compactification of a topological space X is such its extension Y, that Y is a completely regular and countably-compact space, and any closed countably-compact subset of X is closed in Y. But this extension does not always exist. Due to this, the concept of a saturation of a topological space appeared, which is a generalisation of the countably-compactification: instead of the condition of the countably-compactness of Y, it is necessary that any infinite subset of X has a limit point in Y. Meanwhile, the second condition remains unchanged. Such an extension is already defined for any T1-space. In this paper we consider a specific construction of saturation named as Ω-saturation. It is proved that under some additional (necessary and sufficient) condition to the separation of the initial space X, its Ω-saturation is canonically embedded in the Stone – Čech compactification βX. An analogous result is obtained for the countably-compactification by K. Morita.

Author Biographies

Aliaksandr S. Biadrytski, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

master’s degree student at the department of geometry, topology and mathematics teaching methodology, faculty of mechanics and mathematics

Vladimir L. Timokhovich, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

PhD (physics and mathematics), docent; associate professor at the department of geometry, topology and mathematics teaching methodology, faculty of mechanics and mathematics

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Published
2022-04-01
Keywords: saturation of a topological space, countably-compactification in the sense of Morita, Ω-saturation of a topological space, Wallman compactification, ∆-base, Stone – Čech compactification
How to Cite
Biadrytski, A. S., & Timokhovich, V. L. (2022). On the embedding of the Ω-saturation of a topological space. Journal of the Belarusian State University. Mathematics and Informatics, 1, 21-25. https://doi.org/10.33581/2520-6508-2022-1-21-25
Issue
No 1 (2022): Journal of the Belarusian State University. Mathematics and Informatics
Section
Geometry and Topology

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