On countably \(\mu\)-paracompact spaces
The contents of the paper are constructed in the background of generalized topological spaces \((X, \mu)\) (GTS in short) as introduced by Á. Császár. For a GTS \((X, \mu)\), the usual separation axioms are delineated in the usual fashion; \(\mu\)-cover, \(\mu\)-locally finite cover, \(\mu\)-point finite cover etc. are duly introduced for the initiation of countably \(\mu\)-paracompact and countably metacompact spaces. These concepts, in the background of GTSs, however, create a big departure from what happens in usual topological spaces; in fact, a necessary and sufficient condition for a topological space \((X, \tau)\) to be countably paracompact (countably metacompact) is that, for any decreasing sequence \(F_1\supset F_2\supset\cdots\) of nonempty closed subsets of \(X\) with empty intersection, there exists a sequence \(G_1,G_2,\dots\) of open subsets of \(X\) with \(F_{i}\subset G_{i}\) for every \(i=1,2,\dots\) such that \(\bigcap\limits_{i=1}^{\infty} \bar{G_{i}}=\phi\) \((\bigcap\limits_{i=1}^{\infty} {G_{i}}=\phi)\). To establish the departure, the authors construct an example which, as a GTS, is \(\mu T_{4}\) and not countably \(\mu\)-metacompact (a fortiori, not countably \(\mu\)-paracompact, since every countably \(\mu\)-paracompact space is countably \(\mu\)-metacompact) but satisfies the following condition: For any decreasing sequence \(F_1\supset F_2\supset\cdots\) of nonempty \(\mu\)-closed subsets of \(X\) with empty intersection, there exists a sequence \(G_1, G_2, \dots\) of \(\mu\)-open subsets of \(X\) with \(F_{i}\subset G_{i}\) for every \(i=1,2,\dots\) and \(\bigcap\limits_{i=1}^{\infty} \bar{G_{i}}=\phi\); the condition as above is termed condition \((A^{\ast})\). However, it is shown that every \(\mu\)-normal countably \(\mu\)-metacompact space \(X\) satisfies condition \((A^{\ast})\).
scientific article; zbMATH DE number 6806064