Contents

topos theory

# 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 $X$: $\cup_i U_i = X$.

Often it is also required that every point $x \in X$ is in the interior of one of the $U_i$ (e.g in Floyd 1957, but e.g. not in Karoubi & Weibel 2016).

## 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

Applications of closed covers in Čech homology:

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 $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Last revised on June 29, 2022 at 15:42:50. See the history of this page for a list of all contributions to it.