Merotopological spaces
A triple \(X = (X, \tau,\xi)\) is called a merotopological space if it satisfies the following axioms. \((M1)\) \((X,\tau)\) is a topological space; (M2) \((X, \xi)\) is a merotopic space in the sense of \textit{M. Katětov}, where \(\xi\) is the collection consisting of all collections of subsets of \(X\) which are near [Proc. Int. Congr. Math. 1962, 473--479 (1963; Zbl 0116.14105)]; (M3) whenever \(\{\text{cl}_\tau A \mid A \in {\mathcal A}\}\in \xi\), then \(\{A\mid A\in {\mathcal A}\}\in\xi\). It is shown that by defining morphisms in a natural way a topological category \textbf{MerTop} is obtained which contains the categories \textbf{Top} of (all!) topological spaces and \textbf{Near} of nearness spaces as fully embedded subcategories. \textbf{Near} is bireflective in \textbf{MerTop}. Whereas it is still an open problem to characterize internally the spaces belonging to the epireflective hull of the fully embedded subcategory \textbf{TopS} of symmetric \((= R_0)\) topological spaces in \textbf{Near}, it is shown that the the epireflective hull of \textbf{Top} in \textbf{MerTop} is identical with \textbf{MerTop}. The main tool for proving this interesting theorem is the construction of a so-called bunch completion \(X^+\) for each merotopological space \(X\). It turns out that, for regular nearness spaces \(X\), the strict completion \(X^*\) constructed by \textit{H. Herrlich} in [General Topology Appl. 4, 191--212 (1974; Zbl 0288.54004)] is a retract of \(X^+\). Moreover, it is shown that every topological space \((X,\tau)\) can be turned in a natural way into a merotopological space \(X = (X,\tau,\xi)\) such that the Wyler prime closed filter compactification \(\lambda X\) of \((X,\tau)\) is isomorphic to a subspace of \(X^+\). Finally, a merotopological version of the classical Taĭmanov extension theorem is proved which contains the original extension theorem for topological spaces (without any restriction!) as a special case.
Wyler prime closed filter compactification
scientific article; zbMATH DE number 2069147