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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 .

Information

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

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