On the preservation of Baire category under preimages

Author:
Dominikus Noll

Journal:
Proc. Amer. Math. Soc. **107** (1989), 847-854

MSC:
Primary 54C10; Secondary 46A30

DOI:
https://doi.org/10.1090/S0002-9939-1989-0982407-2

MathSciNet review:
982407

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 of the function 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 of this kind defined on a Čech complete space.

**[E]**R. Engelking,*General topology*, Warszawa, 1976.**[F]**Z. Frolík,*Generalizations of the**-property of complete metric spaces*, Czech. Math. J.**10**(85) (1960), 359-378.**[F]**-,*Remarks concerning the invariance of Baire spaces under mappings*, Czech. Math. J.**11**(86), (1961), 381-385. MR**0133098 (24:A2932)****[HM]**R. C. Haworth and R. A. McCoy,*Baire spaces*, Diss. Math. CXLI.**[K]**G. Köthe,*Topological vector spaces*II, Springer-Verlag, New York, 1979. MR**551623 (81g:46001)****[Ne]**T. Neubrunn,*A note on mappings of Baire spaces*, Math. Slovaca**27**(1977), 173-176. Erratum: Math. Slovaca**27**(1977), 442. MR**0454910 (56:13153)****[N]**D. Noll,*Open mapping theorems in topological spaces*, Czech. Math. J.**35**(110), (1985), 373-384. MR**803032 (87a:54013)****[N]**-,*Baire spaces and graph theorems*, Proc. Amer. Math. Soc.**96**, (1986), 141-151. MR**813827 (87e:54072)****[PS]**Z. Piotrowski and A. Szymanski,*Some remarks on category in topological spaces*, Proc. Amer. Math. Soc.**101**(1987), 156-160. MR**897088 (88g:54007)****[T]**F. Topsoe,*Topological games and Čech completeness*. General Topology and its Relations to modern Analysis and Algebra V, Fifth Prague Top. Symp., 1981, pp. 613-630. MR**698466 (84g:54027)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54C10,
46A30

Retrieve articles in all journals with MSC: 54C10, 46A30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0982407-2

Keywords:
Fibre-completeness,
(nearly) feebly open,
(nearly) feebly continuous,
Čech completeness,
open mapping theorem,
preimages of Baire spaces

Article copyright:
© Copyright 1989
American Mathematical Society