New generalized topologies on generalized topological spaces due to Császár (Q653861)

Let \((X, \tau)\) be a topological space and \(\mu\) be one of the following families of sets: pre-open sets, semi-open sets, \(b\)-open sets, \(\alpha\)-open sets, \(\beta\)-open sets of \((X, \tau)\). Then \((X, \mu)\) can be considered as a generalized topological space, i.e., \(\mu\) satisfies that (1) \(\emptyset \in \mu\), and (2) every union of members of \(\mu\) is also a member of \(\mu\). In this paper, a \(\wedge_{\mu}\)-set of a generalized topological space \((X, \mu)\) is introduced. A subset of \(X\) is a \(\wedge _{\mu}\)-set if it can be written as the intersection of a subfamily of \(\mu\). By using this notion, the authors unify the theory of \(\wedge\)-sets, \(\wedge_{\delta}\)-sets, \(\wedge_s\)-sets, etc. Furthermore, \(\vee_{\mu}\)-sets, \(g. \wedge _{\mu}\)-sets and \(g. \vee_{\mu}\)-sets are introduced and studied.