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On the bitopological extension of the Bing metrization theorem. (English) Zbl 0648.54027

All topological spaces considered are \(T_ 1\)-spaces. A pair open cover in a bitopological space (X,\({\mathcal P},{\mathcal Q})\) is a family of pairs \(\{(G_{\alpha},H_{\alpha}):\alpha\in I\}\) such that \(G_{\alpha}\) is \({\mathcal P}\)-open and \(H_{\alpha}\) is \({\mathcal Q}\)-open for each \(\alpha\in I\) and such that \(\{G_{\alpha}\cap H_{\alpha}:\alpha\in I\}\) is a cover of X. The authors call a bitopological space (X,\({\mathcal P},{\mathcal Q})\) pairwise paracompact, if, given a pair open cover (\({\mathcal G},{\mathcal H})\) of (X,\({\mathcal P},{\mathcal Q})\), for each \(x\in X\) there is a sequence \(\{U_ n(x):\) \(n\in N\}\) of \({\mathcal P}\)-neighborhoods of x and a sequence \(\{V_ n(x):\) \(n\in N\}\) of \({\mathcal Q}\)-neighborhoods of x such that (i) \(y\in U_ n(x)\) if and only if \(x\in V_ n(y)\), (ii) if \(x\in X\) there are an \(n\in N\) and a pair \((G_{\alpha},H_{\alpha})\in ({\mathcal G},{\mathcal H})\) with \(U\) \(2_ n(x)\subseteq G_{\alpha}\) and \(V\) \(2_ n(x)\subseteq H_{\alpha}\) [compare e.g. Pac. J. Math. 71, 419-428 (1977; Zbl 0361.54012)]. Note that a pairwise paracompact space (X,\({\mathcal P},{\mathcal Q})\) is pairwise regular, since \({\mathcal P}\) and \({\mathcal Q}\) are \(T_ 1\)-topologies.
Among other things the authors show that a bitopological space is quasi- metrizable if, and only if, it is pairwise developable and pairwise paracompact. (They remark that their definition of pairwise developability is equivalent to the notion of bidevelopability introduced by L. M. Brown [Karadeniz Univ. Math. J. 5, 142-149 (1982; Zbl 0542.54029)].)
{Reviewer’s remark on a problem posed by the authors: If \((X,{\mathcal P},{\mathcal Q})\) is pairwise paracompact, then \({\mathcal P}\vee {\mathcal Q}\) is paracompact. Hence each pairwise paracompact space is \(\delta\)-pairwise paracompact in the sense of T. G. Raghavan and I. L. Reilly [J. Austral. Math. Soc. (Series A) 41, 268-274 (1986; Zbl 0609.54023)]. On the other hand if \((X,{\mathcal P})\) is a Moore space that is not quasi- metrizable and \({\mathcal D}\) is the discrete topology on X, then \((X,{\mathcal P},{\mathcal D})\) is \(\delta\)-pairwise paracompact, but not pairwise paracompact, since \({\mathcal P}\) is not \(\gamma\)-refinable [see P. Fletcher and W. F. Lindgren, Quasi-uniform spaces (1982; Zbl 0501.54018) for the necessary definitions and facts]. The following example shows that Theorem 3 is not correct. Let \(X=\omega \times \{0,1\}\). Assume that all points of X except \(x\in \{(0,0),(0,1)\}\) (except \(x\in \{(1,0),(1,1)\})\) are \({\mathcal P}\)-isolated (\({\mathcal Q}\)- isolated). For an exceptional point x a basic \({\mathcal P}\)-neighborhood (\({\mathcal Q}\)-neighborhood) is \(\{x\}\cup [(\omega \times \{0\}\setminus F]\) ({\(x\}\cup [(\omega \times \{1\})\setminus F])\) where F is a finite subset of X. The projection from X onto \(\omega\) (equipped with the quotient topologies \({\mathcal P}'\) and \({\mathcal Q}')\) is a continuous open 2- to one mapping such that \((\omega,{\mathcal P}',{\mathcal Q}'\) is not pairwise regular, although \((X,{\mathcal P},{\mathcal Q})\) is quasi-metrizable.}
Reviewer: H.-P.Künzi

MSC:

54E55 Bitopologies
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E30 Moore spaces
54E35 Metric spaces, metrizability
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