# nLab closed cover

topos theory

## Theorems

#### Topology

topology

algebraic topology

# Contents

## Definition

A closed cover of a topological space $X$ is a collection $\{U_i \subset X\}$ of closed subsets of $X$ whose union equals $X$: $\cup_i U_i = X$. Usually it is also required that every point $x \in X$ is in the interior of one of the $U_i$.

## 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)