On the preservation of Baire category under preimages

Noll, Dominikus

doi:10.1090/S0002-9939-1989-0982407-2

On the preservation of Baire category under preimages
HTML articles powered by AMS MathViewer

by Dominikus Noll PDF

Proc. Amer. Math. Soc. 107 (1989), 847-854 Request permission

Abstract:

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.

References

General topology

Generalizations of the

-property of complete metric spaces

10

Zdeněk Frolík, Remarks concerning the invariance of Baire spaces under mappings, Czechoslovak Math. J. 11(86) (1961), 381–385 (English, with Russian summary). MR 133098

Baire spaces

Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 237, Springer-Verlag, New York-Berlin, 1979. MR 551623
Tibor Neubrunn, A note on mappings of Baire spaces, Math. Slovaca 27 (1977), no. 2, 173–176 (English, with Russian summary). MR 454910
Dominik Noll, Open mapping theorems in topological spaces, Czechoslovak Math. J. 35(110) (1985), no. 3, 373–384. MR 803032
Dominik Noll, Baire spaces and graph theorems, Proc. Amer. Math. Soc. 96 (1986), no. 1, 141–151. MR 813827, DOI 10.1090/S0002-9939-1986-0813827-1
Zbigniew Piotrowski and Andrzej Szymański, Some remarks on category in topological spaces, Proc. Amer. Math. Soc. 101 (1987), no. 1, 156–160. MR 897088, DOI 10.1090/S0002-9939-1987-0897088-4
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. MR 698466