nLab
closed cover

Contents

Definition

A closed cover of a topological space XX is a collection {U iX}\{U_i \subset X\} of closed subsets of XX whose union equals XX: iU i=X\cup_i U_i = X. Usually it is also required that every point xXx \in X is in the interior of one of the U iU_i.

Properties

Closed covers can be obtained from open covers by forming the closure of each of the open subsets. The result clearly satisfies the clause that every point is in the interior of one of the closed subsets.

References

General

  • Dragan Janković, Chariklia Konstadilaki, On covering properties by regular closed sets, Mathematica Pannonica, 7/1 (1996) 97-111 (pdf)

Applications of closed covers in Čech homology is discussed in

  • E. Floyd, Closed coverings in Čech homology theory (pdf)

Related discussion is also in this MO thread

Examples

In analytic geometry, affinoid domains have closed sets as analytic spectra and hence the topological space underlying a Berkovich analytic spaces is equipped by a closed cover by affioid domains

  • Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on pp-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Revised on July 17, 2014 14:44:38 by Urs Schreiber (82.136.246.44)