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On relative (bi)topological properties and dimension functions. (English) Zbl 1172.54021

In this rather large paper (126 pages) a systematic study of relative properties of bitopological spaces is presented. At first, in part one (Sections 2–6), special attention is given to relative separation axioms and relative connectedness. Next, in part two (Sections 7–12), the author devotes his study, on the one hand, to relative bitopological inductive, separately inductive and covering dimension functions and, on the other hand, to relative versions of Baire spaces.
The leading idea of the paper is the following: Let \({\mathcal P}\) be a bitopological property, let \(X\) be a bitopological space and let \(Y\) be a bitopological subspace of \(X\); define “\(Y\) has property \({\mathcal P}\) relative to \(X\)” in such a way that when \(Y=X\) we obtain “\(X\) has property \({\mathcal P}\)”. For example, let \((X,\tau_1,\tau_2)\) be a bitopological space and let \((Y,\tau'_1,\tau'_2)\) be a bitopological subspace of \(X\). “\(Y\) is \((i,j)\)-regular in \(X\) (\(i,j\in\{1,2\}\), \(i\neq j\))” means that for each \(y\in Y\) and each \(\tau_i\)-closed set \(F\) with \(y\notin F\), there are disjoint sets \(U\in\tau_i\) and \(V\in \tau_j\) such that \(y\in U\) and \(F\cap Y\subseteq V\).
This is the extension to the bitopological setting of an idea of A. V. Arhangel’skii in [Sci. Math. Jpn. 55, No.1, 153–201 (2001; Zbl 0994.54024)].
Among the large list of properties considered are the following: separation axioms (\(T_0\), \(T_1\), \(T_2\), Tychonoff, regular, and normal), connectedness, covering properties (compactness, Lindelöf and paracompactness), dimension functions and the Baire property.

MSC:

54E55 Bitopologies
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54F45 Dimension theory in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54C05 Continuous maps
54E52 Baire category, Baire spaces

Citations:

Zbl 0994.54024
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