Geodesic
Open and Proper Maps Characterized by Continuous Setvalued Maps
Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 929-936

Voir la notice de l'article provenant de la source Cambridge University Press

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 . 
Colebunders, Eva Lowen-. Open and Proper Maps Characterized by Continuous Setvalued Maps. Canadian journal of mathematics, Tome 33 (1981) no. 4, pp. 929-936. doi: 10.4153/CJM-1981-073-6
@article{10_4153_CJM_1981_073_6,
     author = {Colebunders, Eva Lowen-},
     title = {Open and {Proper} {Maps} {Characterized} by {Continuous} {Setvalued} {Maps}},
     journal = {Canadian journal of mathematics},
     pages = {929--936},
     year = {1981},
     volume = {33},
     number = {4},
     doi = {10.4153/CJM-1981-073-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-073-6/}
}
TY  - JOUR
AU  - Colebunders, Eva Lowen-
TI  - Open and Proper Maps Characterized by Continuous Setvalued Maps
JO  - Canadian journal of mathematics
PY  - 1981
SP  - 929
EP  - 936
VL  - 33
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-073-6/
DO  - 10.4153/CJM-1981-073-6
ID  - 10_4153_CJM_1981_073_6
ER  - 
%0 Journal Article
%A Colebunders, Eva Lowen-
%T Open and Proper Maps Characterized by Continuous Setvalued Maps
%J Canadian journal of mathematics
%D 1981
%P 929-936
%V 33
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-073-6/
%R 10.4153/CJM-1981-073-6
%F 10_4153_CJM_1981_073_6

[1] 1. Bourbaki, N., Topologie générale, 4th éd., Eléments de Mathématique, Fas II (Hermann Paris, 1965). Google Scholar

[2] 2. Choquet, G., Convergences, Ann. Univ. Grenoble Sect. Sci. Math. Phys. 23 (1948), 57–112. Google Scholar

[3] 3. Fischer, H., Limesraume, Math. Ann. 137 (1959), 269–303. Google Scholar

[4] 4. Frolik, Z., Concerning topological convergence of sets, Czechoslovak Math. J. 10 (1960), 168–180. Google Scholar

[5] 5. Gàhler, W., Beitrâge zur théorie der Limesraume, Theory of sets and topology, Berlin (1972), 161–197. Google Scholar

[6] 6. Hausdorff, F., Mengenlehre (Berlin-Leipzig, 1927). Google Scholar

[7] 7. Kent, D. and Richardson, G., Open and proper maps between convergence spaces, Czechoslovak Math. J. 23 (1973), 15–23. Google Scholar

[8] 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] 9. Lowen-Colebunders, E., The Choquet hyperspace structure for convergence spaces, Math. Nachr. 95 (1980), 17–26. Google Scholar

[10] 10. Mrowka, S., On the convergence of nets of sets, Fund. Math. 45 (1958), 237–246. Google Scholar

Cité par Sources :