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Baire isomorphisms at the first level and dimension. (English) Zbl 0960.54024

Summary: For a topological space \(X\) let \({\mathcal Z}_\sigma(X)\) denote the family of subsets of \(X\) which can be represented as a union of countably many zero-sets. A bijection \(h:X\to Y\) between topological spaces \(X\) and \(Y\) is a first level Baire isomorphism if \(f(Z)\in {\mathcal Z}_\sigma(Y)\) and \(f^{-1}(Z') \in{\mathcal Z}_\sigma (X)\) whenever \(Z\in{\mathcal Z}_\sigma(X)\) and \(Z'\in {\mathcal Z}_\sigma (Y)\). A space is \(\sigma\)-(pseudo)compact if it can be represented as the union of a countable family consisting of its (pseudo)compact subsets. Generalizing results of J. E. Jayne and C. A. Rogers [Mathematika 26, 125-156 (1979; Zbl 0443.54029)] and of A. Chigogidze [Commentat. Math. Univ. Carol. 26, 811-820 (1985; Zbl 0587.54030)] we show that first level Baire isomorphic, \(\sigma\)-pseudocompact (in particular, \(\sigma\)-compact) Tikhonov spaces have the same covering dimension dim.

MSC:

54F45 Dimension theory in general topology
54D45 Local compactness, \(\sigma\)-compactness
54C99 Maps and general types of topological spaces defined by maps
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
54E45 Compact (locally compact) metric spaces
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