$n$-H-closed spaces

TL;DR

This paper generalizes the concept of H-closed spaces to a broader class called n-H-closed spaces, extending known constructions like the Katětov extension to non-Hausdorff settings.

Contribution

It introduces the notion of n-H-closed spaces and generalizes existing H-closed extensions to non-Hausdorff spaces, including the construction of a maximal n-H-closed extension.

Findings

Defined n-H-closed spaces and their properties.

Extended Katětov H-closed extension to n-H-closed spaces.

Provided a construction for the maximal n-H-closed extension.

Abstract

In this paper we extend the theory of H-closed extensions of Hausdorff spaces to a class of non-Hausdorff spaces, defined in \cite{B}, called $n$-Hausdorff spaces. The notion of H-closed is generalized to an $n$-H-closed space. Known construction for Hausdorff spaces $X$, such as the Kat\v{e}tov H-closed extension $\kappa X$, are generalized to a maximal $n$-H-closed extension denoted by $n$-$\kappa X$.