×

Mappings which are global homeomorphisms. (English. Russian original) Zbl 0614.58008

Math. Notes 39, 142-145 (1986); translation from Mat. Zametki 39, No. 2, 260-267 (1986).
Let X and Y be connected infinite-dimensional Fréchet manifolds, where Y is simply connected, and let \(f: X\to Y\) be a continuous map which is assumed to be either nonconstant and closed, or proper. Let \(P_ f\) be a closed subset of X such that at each point of \(X/P_ f\) the map f is not a local homeomorphism. Assume that \(P_ f\) is an at most countable union of compact sets and that f is open. Then f is a homeomorphism of X onto Y.
Reviewer: D.S.Janković

MSC:

58C07 Continuity properties of mappings on manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Banach and S. Mazur, ?Über mehrdeutige stetige Abbildungen,? Stud. Math.,5, 174-178 (1934). · Zbl 0013.08202
[2] P. Church and E. Hemmingsen, ?Light open maps on n-manifolds,? Duke Math. J.,27, 527-536 (1960). · Zbl 0117.40502 · doi:10.1215/S0012-7094-60-02750-2
[3] S. Smale, ?An infinite-dimensional version of Sard’s theorem,? Am. J. Math.,87, No. 4, 861-866 (1965). · Zbl 0143.35301 · doi:10.2307/2373250
[4] M. Berger and R. Plastock, ?On the singularities of nonlinear Fredholm operators,? Bull. Am. Math. Soc.,83, No. 6, 1316-1318 (1977). · Zbl 0382.47032 · doi:10.1090/S0002-9904-1977-14431-5
[5] R. Plastock, ?Nonlinear Fredholm maps of index zero and their singularities,? Proc. Am. Math. Soc.,68, 317-322 (1978). · Zbl 0392.47035 · doi:10.1090/S0002-9939-1978-0464283-2
[6] M. Berger and R. Plastock, ?On the singularities of nonlinear Fredholm operators of positive index,? Proc. Am. Math. Soc.,79, No. 2, 217-221 (1980). · Zbl 0444.47044 · doi:10.1090/S0002-9939-1980-0565342-5
[7] E. Michael, ?Local properties of topological spaces,? Duke Math. J.,21, 163-172 (1954). · Zbl 0055.16203 · doi:10.1215/S0012-7094-54-02117-1
[8] E. Michael, ?A note on closed maps and compact sets,? Israel J. Math., Sec. F,2, No. 3, 173-176 (1964). · Zbl 0136.19303 · doi:10.1007/BF02759940
[9] W. Cutler, ?Negligible subsets of infinite dimensional Frechet manifolds,? Proc Am. Math. Soc.,23, 668-675 (1969). · Zbl 0195.53603
[10] M. Henriksen and J. Isbell, ?Some properties of compactifications,? Duke Math. J.,25, No. 1, 83-106 (1958). · Zbl 0081.38604 · doi:10.1215/S0012-7094-58-02509-2
[11] C. Ho, ?A note on proper maps,? Proc. Am. Math. Soc.,51, No. 1, 237-241 (1975). · Zbl 0273.54007 · doi:10.1090/S0002-9939-1975-0370471-3
[12] H. Cartan, Differential Forms, Wiley.
[13] R. S. Sadyrkhanov, ?Tests for finite multiplicity of covering maps,? Dokl. Akad. Nauk SSSR,273, No. 1, 54-58 (1983). · Zbl 0558.54011
[14] R. S. Sadyrkhanov, ?Condition for a map to be a local homeomorphism,? Dokl. Akad. Nauk SSSR,275, No. 6, 1316-1320 (1984). · Zbl 0594.54012
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.