Paracompactness and full normality in ditopological texture spaces
The authors consider the following bitopological-like structure. Let \(S\) be a set and let \({\mathcal S}\subseteq P(S)\) be a complete and completely distributive lattice (with \(\emptyset\) and \(S\) as bounds) in which finite joins and arbitrary meets are those of \((P(S),\subseteq)\) and which satisfies a \(T_1\)-type separation axiom. Then \((\tau,\kappa)\) is called a ditopology on \((S, {\mathcal S})\) if \(\tau\subseteq{\mathcal S}\) is closed under finite infs and arbitrary sups and \(\kappa \subseteq {\mathcal S}\) is closed under finite sups and arbitrary infs. Furthermore, a family \(\{(G_\alpha, F_\alpha): \alpha\in A\} \subseteq {\mathcal S}\times {\mathcal S}\) is called a dicover of \(S\) if \(\bigcap\{F_\alpha: \alpha\in A_1\} \subseteq \bigvee\{G_\alpha: \alpha\in A_2\}\) for any partitions \(A_1\) and \(A_2\) of \(A\). The concept of a dicover is then used to define notions of paracompactness and full normality.
scientific article; zbMATH DE number 1236167
scientific article