Construction of the Katetov Extension of a Hausdorff Space

Year 2021, Volume: 9 Issue: 1 , 159 - 163 , 28.04.2021

Marco Mpimbo , Mayila Shega

https://izlik.org/JA37PF96WJ

Abstract

Katetov extension $\kappa X$ of Hausdorff space $X$ has been studied extensively as the largest H-closed extension of a Hausdorff space. Recall that, a Hausdorff space $X$ is said to be an H-closed space if it is closed in every Hausdorff space in which it is embedded. Although Kat\v{e}tov extensions of Hausdorff spaces have been extensively studied, to date there has been very little work on either its construction or its structure (topology). In this paper, we give the detailed algorithm for constructing such a space by using filters on $X$. The basis generating the topology on $\kappa X$ contains the open sets of the form $V\cup\{\Gamma: V\in\Gamma\in \kappa X-X\}$ or $U\subset X$ where both $U$ and $V$ are open subsets of $X$ and $\Gamma$ is a non-convergent ultra-filter on $X$ containing $V$. Moreover, using simple approach, it is proved that Kat\v{e}tov extension $\kappa X$ is a Hausdorff space, H-closed, maximal and unique extension for $X$.

Keywords

Extension , Hausdorff spaces , H-closed spaces

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