Generalized open sets
Various kinds of generalized open sets are studied together in this article. The author considers monotone mappings \(\gamma:\exp X\to \exp X\) (monotone in the sense that \(A\subset B\) implies \(\gamma(a)\subset \gamma(B)\)), and develops properties common to several classes of generalized open sets. A subset \(A\) of \(X\) is called \(\gamma\)-open iff \(A\subset\gamma A\), and, it is \(\gamma\)-closed iff \(X-A\) is \(\gamma\)-open. If \(X\) is a topological space and \(i\) is the interior map (i.e., \(iA=\text{int} A\)), then the \(i\)-open sets are the open sets in the given topology. Definitions of generalized open sets can then be given in terms of the appropriate \(\gamma\). The preopen sets of \textit{A. S. Mashhour, M. E. Abd El-Monsef}, and \textit{S. N. El-Deep} [Proc. Math. Phys. Soc. Egypt. 53, 47-53 (1982; Zbl 0571.54011)] become the \(\gamma\)-open sets for \(\gamma=ic\), where \(c\) is the closure map; the semi-open sets of \textit{N. Levine} [Am. Math. Mon. 70, 36-41 (1963; Zbl 0113.16304)] are the \(\gamma\)-open sets for \(\gamma=ici\). Other generalizations such as \(\alpha\)-open and \(\beta\)-open can be couched in analogous terms. Theorems involving \(\gamma\)-open sets will then apply automatically to each of these concepts; i.e., Proposition 1.1: Any union of \(\gamma\)-open sets is \(\gamma\)-open. The author continues with studies of intersections of \(\gamma\)-open sets, \(\gamma\)-open sets in subspaces, locally \(\gamma\)-closed sets, and \(\gamma\)-regular sets, with the natural definitions of these concepts.
scientific article; zbMATH DE number 1354993