One-Point λ-Compactification via Grills

Abstract

A space $Y$ is called an extension of a space $X$ if $Y$ contains $X$ as a dense subspace. An extension $Y$ of $X$ is called a one-point extension of $X$ if $Y\backslash X$ is a singleton. In present paper, we extend the well-known and important fact “any locally compact Hausdorff non-compact space $X$ has a one-point compact Hausdorff extension”, which was proved by Alexandroff, to more general topological space.