Shakhmatov, Dmitri B. Baire isomorphisms at the first level and dimension. (English) Zbl 0960.54024 Topology Appl. 107, No. 1-2, 153-159 (2000). 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. Cited in 1 Document 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 Keywords:\(\sigma\)-compact; Borel set; pseudo-compact; Baire set; first level Borel isomorphism; factorization; first level Baire isomorphism; covering dimension Citations:Zbl 0443.54029; Zbl 0587.54030 PDFBibTeX XMLCite \textit{D. B. Shakhmatov}, Topology Appl. 107, No. 1--2, 153--159 (2000; Zbl 0960.54024) Full Text: DOI