## 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

### Citations:

Zbl 0443.54029; Zbl 0587.54030
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