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\(F_\sigma\)-mappings between paracompact perfect spaces. (English) Zbl 1470.54023

The main result proves that a mapping \(f:X\to Y\) is piecewise closed (i.e., \(f\) can be covered by continuous closed \(f_n:X_n\to Y\)) if \(f\) is an \(F_\sigma\)-mapping (i.e., \(f(E)\) and \(f^{-1}(F)\) are \(F_\sigma\) for \(E,F\) of type \(F_\sigma\) in the corresponding space) and \(f^{-1}(y)\) are compact for \(y\in Y\).
This result was proved by R. W. Hansell, J. E. Jayne and C. A. Rogers [Mathematika 32, 229–247 (1985; Zbl 0618.54015)] for the case of absolute Souslin-\(\mathcal F\) spaces \(X\) and \(Y\). Here the spaces \(X\) and \(Y\) are assumed to be Souslin-\(\mathcal F\) in some regular completely Baire spaces which are moreover first-countable, paracompact, and perfect.
A generalization of former results of Jayne and Rogers, and also of results of M. Kačena et al. [Real Anal. Exch. 38, No. 1, 121–132 (2013; Zbl 1272.03149)], was proved by the author in [Acta Math. Hung. 155, No. 2, 406–415 (2018; Zbl 1413.54093)] and is used to reduce the proof to the case of continuous \(F_\sigma\)-mappings. Then constructions proving the existence of the closed mappings \(f_n\) are carried over.

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E52 Baire category, Baire spaces
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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References:

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