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Compactness for a class of hit-and-miss hyperspaces

  • René Bartsch
  • Harry Poppe
Article
  • 36 Downloads

Abstract

In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space (X, τ), namely the so called Δ-hit-and-miss-topologies with Δ⊇Cl(X) (or Δ-topologies), which was initiated by the second author in 1965, it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of (X, τ) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.

Keywords

Topological Space Compact Subset Closed Subset Open Cover Usual Notion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer 2002

Authors and Affiliations

  1. 1.Universität Rostock Fachbereich InformatikRostockGermany
  2. 2.Universität Rostock Fachbereich MathematikRostockGermany

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