One-point extensions and local topological properties (Q2847591)

Suppose \(\mathcal P\) is a topological property such that (1) \(\mathcal P\) is hereditary to closed sets, (2) any space \(X\) has \(\mathcal P\) if \(X\) is a finite union of closed subspaces with \(\mathcal P\), and (3) (the Mrówka condition) if \(X\) is a completely regular space and there is a point \(p \in X\) with a basis \(\mathcal B\) of open neighborhoods such that \(X-B\) has \(\mathcal P\) for every \(B \in \mathcal{B}\), then \(X\) has \(\mathcal P\). Suppose \(\mathcal Q\) is a topological property which implies complete regularity and satisfies (1) and (3) above. The author shows that any locally-\(\mathcal P\) non-\(\mathcal P\) space with \(\mathcal Q\) has a one-point extension satisfying both \(\mathcal P\) and \(\mathcal Q\). This answers a question posed by \textit{S. Mrówka} and \textit{J. H. Tsai} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 19, 1035--1040 (1971; Zbl 0224.54034)].