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Čech-complete spaces and the upper topology

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
70
Identifiers
DOI: 10.1016/0166-8641(95)00090-9
Keywords
  • Compact-Covering Map
  • Čech-Complete Space
  • Lindelöf Space
  • Hyperspace
  • Vietoris Topology
  • Upper Topology
Disciplines
  • Mathematics

Abstract

Abstract Let X be a topological space and let K(X) be the set of all compact subsets of X. The purpose of this note is to prove the following: if X is regular and q-space, then X is Lindelöf and Čech-complete if and only if there exists a continuous map f from a Lindelöf and Čech-complete space Y to the space K(X) endowed with the upper topology, such that f( Y) is cofinal in ( K(X), ⊂) . This result extends the following result of Saint Raymond and Christensen: if X is separable metrizable, then X is a Polish space if and only if the space K(X) endowed with the Vietoris topology is the continuous image of a Polish space.

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