Medvedev, Sergey \(F_\sigma\)-mappings between paracompact perfect spaces. (English) Zbl 1470.54023 Topology Appl. 281, Article ID 107208, 11 p. (2020). 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. Reviewer: Petr Holický (Praha) 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.) Keywords:Souslin-\(\mathcal F\) set; \(F_\sigma\)-mapping; piecewise closed mapping; completely Baire space; paracompact perfect space Citations:Zbl 1413.54093; Zbl 0618.54015; Zbl 1272.03149 PDFBibTeX XMLCite \textit{S. Medvedev}, Topology Appl. 281, Article ID 107208, 11 p. (2020; Zbl 1470.54023) Full Text: DOI References: [1] Banakh, T.; Bokalo, B., On scatteredly continuous maps between topological spaces, Topol. Appl., 157, 108-122 (2010) · Zbl 1196.54021 [2] Engelking, R., General Topology (1977), PWN: PWN Warszawa [3] Jayne, J. E.; Rogers, C. A., Fonctionc fermées en partie, C. R. Acad. Sci. Paris Sér. A, 291, 667-670 (1980) · Zbl 0463.54036 [4] Jayne, J. E.; Rogers, C. A., Piece-wise closed functions, Math. Ann.. Math. Ann., Math. Ann., 267, 143-518 (1984), and Corrigendum: · Zbl 0539.54006 [5] Jayne, J. E.; Rogers, C. A., First level Borel functions and isomorphisms, J. Math. Pures Appl., 61, 177-205 (1982) · Zbl 0514.54026 [6] Hansell, R. W.; Jayne, J. E.; Rogers, C. A., Piece-wise closed functions and almost discretely σ-decomposable families, Mathematika, 32, 229-247 (1985) · Zbl 0618.54015 [7] Holický, P.; Spurný, J., \( F_\sigma \)-mappings and the invariance of absolute Borel classes, Fundam. Math., 182, 193-204 (2004) · Zbl 1059.54025 [8] Kačena, M.; Motto Ros, L.; Semmes, B., Some observations on “A new proof of a theorem of Jayne and Rogers”, Real Anal. Exch., 38, 1, 121-132 (2012/2013) · Zbl 1272.03149 [9] Medvedev, S. V., On piecewise continuous mappings of paracompact spaces, Sib. Electron. Math. Rep., 15, 214-222 (2018) · Zbl 1396.54016 [10] Medvedev, S. V., On the Jayne-Rogers theorem, Acta Math. Hung., 155, 2, 406-415 (2018) · Zbl 1413.54093 [11] Sierpiński, W., Sur une propriété topologique des ensembles dénombrables denses en soi, Fundam. Math., 1, 11-16 (1920) · JFM 47.0175.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.