An elementary approach to `algebra \(\cap\) topology = compactness' (Q2563767)

\textit{H. Herrlich} and \textit{G. E. Strecker} characterized the category \({\mathcal C}\)\textit{omp}\(_2\) of compact Hausdorff spaces as the only nontrivial full epireflective subcategory of the category \({\mathcal T}\)\textit{op}\(_2\) of all Hausdorff spaces that is varietal with respect to its underlying set functor [Gen. Topol. Appl. 1, 283-287 (1971; Zbl 0231.18007)]. Their proof requires Noble's theorem, i.e., a space is compact Hausdorff iff every of its powers is normal. In this paper the author points out that Noble's theorem is far from being elementary and shows an elementary solution for the above characterization by making observations for some subcategories of all topological spaces. By the same idea he also gives an elementary approach to \textit{D. Petz}' characterization of the category \({\mathcal C}\)\textit{omp}\(_2\) [Stud. Sci. Math. Hung. 12, 407-408 (1977; Zbl 0448.54023)], i.e., \({\mathcal C}\)\textit{omp}\(_2\) is the only nontrival, full, isomorphism-closed, and epireflective subcategory of \({\mathcal T}\)\textit{op}\(_2\) which is closed under dense extensions and strictly contained in \({\mathcal T}\)\textit{op}\(_2\).