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Open and Proper Maps Characterized by Continuous Setvalued Maps

Published online by Cambridge University Press:  20 November 2018

Eva Lowen- Colebunders
Eva Lowen- Colebunders*
Affiliation:
Vrije Universiteit Brussel, Brussel, Belgium
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In the first part of the paper, given a continuous map f from a Hausdorff topological space X onto a Hausdorff topological space Y, we consider the reciprocal map f* from Y into the collection of closed subsets of X, which maps y ∈ Y to . is endowed with the pseudotopological structure of convergence of closed sets. We will use the filter description of this convergence, as defined by Choquet and Gähler [2], [5], which is equivalent to the “topological convergence” of sets as introduced by Frolík and Mrówka [4], [10]. These notions in fact generalize the convergence of sequences of sets defined by Hausdorff [6]. We show that the continuity of f* is equivalent to the openness of f. On f*(Y), the set of fibers of f, we consider the pseudotopological structure induced by the closed convergence on .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 4 , 01 August 1981 , pp. 929 - 936
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bourbaki, N., Topologie générale, 4th éd., Eléments de Mathématique, Fas II (Hermann Paris, 1965).Google Scholar
2. Choquet, G., Convergences, Ann. Univ. Grenoble Sect. Sci. Math. Phys. 23 (1948), 57–112.Google Scholar
3. Fischer, H., Limesraume, Math. Ann. 137 (1959), 269–303.Google Scholar
4. Frolik, Z., Concerning topological convergence of sets, Czechoslovak Math. J. 10 (1960), 168–180.Google Scholar
5. Gàhler, W., Beitrâge zur théorie der Limesraume, Theory of sets and topology, Berlin (1972), 161–197.Google Scholar
6. Hausdorff, F., Mengenlehre (Berlin-Leipzig, 1927).Google Scholar
7. Kent, D. and Richardson, G., Open and proper maps between convergence spaces, Czechoslovak Math. J. 23 (1973), 15–23.Google Scholar
8. Lowen-Colebunders, E., An internal and an external characterization of convergence spaces in which adhérences of filters are closed, Proc. Amer. Math. Soc. 72 (1978), 205–210.Google Scholar
9. Lowen-Colebunders, E., The Choquet hyperspace structure for convergence spaces, Math. Nachr. 95 (1980), 17–26.Google Scholar
10. Mrowka, S., On the convergence of nets of sets, Fund. Math. 45 (1958), 237–246.Google Scholar