nLab
closed cover

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Topology

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

  • 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

Revised on May 2, 2012 15:56:55 by Urs Schreiber (82.113.99.15)