A uniform approach to normality for topological spaces (Q2800905)

A large amount of research in topology includes semi-open sets, \(\alpha\)-open sets, \(\beta\)-open sets, pre-open sets, etc. Then later on A-sets, B-sets, C-sets, etc. were defined and studied. A. Császár used the class of mappings \(\gamma: P(X)\rightarrow P(X)\) such that \(A\subseteq B \Rightarrow \gamma(A)\subseteq \gamma(B)\), to give an answer that these classes can be obtained by using one common definition. All these families form ``generalized topologies''.NEWLINENEWLINEIn this paper, the authors introduce and investigate a generalized form of normality called \((\lambda,\mu)\)-normality for generalized topologies. They use two GT's simultaneously in their definition. By this definition, they obtain a more general form of normality. They provide a uniform approach towards various notions of normality existing in the literature. For example, let \((X,\tau)\) be a topological space, then by taking \(\lambda = \mu = \mathrm{int}\), they get normality for \(X\); by taking \(\lambda=\mathrm{int}\), \(\mu=cl^{*}_{\theta}\), they get \(\theta\)-normality; by taking \(\lambda=\mathrm{int}\), \(\mu=cl^{*}_{\delta}\), they get \(\triangle\)-normality for \(X\). Moreover, if \((X,\tau_{1}, \tau_{2})\) is a bitopological space by taking \(\lambda=\mathrm{int}_{\tau_{1}}\) and \(\mu= \mathrm{int}_{\tau_{2}}\) they get pairwise normality of \((X,\tau_{1}, \tau_{2})\). Then the authors also give examples that show how different sets of conditions lead to different variants of normality.NEWLINENEWLINEFinally, the authors provide generalized versions of Urysohn's lemma and the Tietze extension theorem, which hold for \((\lambda,\mu)\)-normality.