On the embedding of the $\Omega$-saturation of a topological space

A. S. Biadrytski ; V. L. Timokhovich

Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2022), pp. 21-25

Résumé

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 $T_{1}$-space. In this paper we consider a specific construction of saturation named as $\Omega$-saturation. It is proved that under some additional (necessary and sufficient) condition to the separation of the initial space $X$, its $\Omega$-saturation is canonically embedded in the Stone – Cech compactification $\beta X$. An analogous result is obtained for the countably-compactification by K. Morita.

Détail

Keywords: saturation of a topological space; countably-compactification in the sense of Morita; $\Omega$-saturation of a topological space; Wallman compactification; $\Delta$-base; Stone - Cech compactification.