Construction of the Katetov Extension of a Hausdorff Space

Authors : Marco MPİMBO, Mayila SHEGA

Pages : 159-163

View : 13 | Download : 17

Publication Date : 2021-04-28

Article Type : Research Paper

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 insert ignore into journalissuearticles values(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