ONE-POINT EXTENSIONS AND LOCAL TOPOLOGICAL PROPERTIES

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 \ 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, Mrowka and Tsai ['On local topological properties. II', Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 19 (1971), 1035-1040] posed the following more general question: For what pairs of topological properties P and Q does a locally-P space X having Q possess a one-point extension having both P and Q? Here, we provide an answer to this old question.