One-point extensions and local topological properties

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. P. Alexandroff proved that any locally compact non-compact Hausdorff space $X$ has a one-point compact Hausdorff extension, called the one-point compactification of $X$. Motivated by this, S. Mr\'{o}wka and J.H. Tsai [On local topological properties. II, Bull. Acad. Polon. Sci. S\'{e}r. Sci. Math. Astronom. Phys. 19 (1971), 1035-1040] posed the following more general question: For what pairs of topological properties ${\mathscr P}$ and ${\mathscr Q}$ does a locally-${\mathscr P}$ space $X$ having ${\mathscr Q}$ possess a one-point extension having both ${\mathscr P}$ and ${\mathscr Q}$? Here, we provide an answer to this old question.