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Pairwise weakly Hausdorff spaces. (English) Zbl 1111.54026
A bitopological space \((X,\tau _1,\tau _2)\) is defined to be a pairwise weakly Hausdorff space (denoted by PWT\(_2\)-space) if \(\tau _j\cdot \text{cl}(x)= \tau _i\cdot \text{cl}(y)\) whenever there is a net \(S\) in \(X\) such that \(x\in \tau _i\cdot \lim (S)\) and \(y\in \tau _j \cdot \lim (S)\). The \(\tau _i\)-limit set of \(S\) \(\tau _i \cdot \lim (S)\) is defined to be the collection of points \(x\) such that \(S\tau _i\)-converges to \(x\).
The author proves some basic results about PWT\(_2\)-spaces (e.g. to be productive, hereditary and topological). Some relationships between these spaces and other bitopological spaces satisfying some separation axioms are studied. In the last section of the paper, the author introduces the notion of a \(P\)-paracompact bitopological space and proves some results about such spaces.
Reviewer: Jan Paseka (Brno)
MSC:
54E55 Bitopologies
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D30 Compactness
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