zbMATH — the first resource for mathematics

On the preservation of Baire category under preimages. (English) Zbl 0687.54012
Summary: We discuss the problem of preservation of Baire category under continuous and feebly open preimages. We obtain a solution by imposing a completeness condition on the fibres \(f^{-1}(y)\) of the function f under consideration. Based on a theorem on the invariance of residuality under continuous and nearly feebly open images, we also derive a result concerning the preservation of category under continuous and nearly feebly open preimages. We end up with an open mapping theorem for functions f of this kind defined on a Čech complete space.

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54E52 Baire category, Baire spaces
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
Full Text: DOI
[1] R. Engelking, General topology, Warszawa, 1976. · Zbl 0373.54001
[2] Z. Frolík, Generalizations of the \( {G_\delta }\)-property of complete metric spaces, Czech. Math. J. 10 (85) (1960), 359-378. · Zbl 0100.18701
[3] Zdeněk Frolík, Remarks concerning the invariance of Baire spaces under mappings, Czechoslovak Math. J. 11 (86) (1961), 381 – 385 (English, with Russian summary). · Zbl 0104.17204
[4] R. C. Haworth and R. A. McCoy, Baire spaces, Diss. Math. CXLI. · Zbl 0344.54001
[5] Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 237, Springer-Verlag, New York-Berlin, 1979. · Zbl 0417.46001
[6] Tibor Neubrunn, A note on mappings of Baire spaces, Math. Slovaca 27 (1977), no. 2, 173 – 176 (English, with Russian summary). · Zbl 0371.54023
[7] Dominik Noll, Open mapping theorems in topological spaces, Czechoslovak Math. J. 35(110) (1985), no. 3, 373 – 384. · Zbl 0609.54013
[8] Dominik Noll, Baire spaces and graph theorems, Proc. Amer. Math. Soc. 96 (1986), no. 1, 141 – 151. · Zbl 0593.54012
[9] Zbigniew Piotrowski and Andrzej Szymański, Some remarks on category in topological spaces, Proc. Amer. Math. Soc. 101 (1987), no. 1, 156 – 160. · Zbl 0634.54002
[10] F. Topsøe, Topological games and Čech completeness, General topology and its relations to modern analysis and algebra, V (Prague, 1981) Sigma Ser. Pure Math., vol. 3, Heldermann, Berlin, 1983, pp. 613 – 630.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.