nLab closed cover

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.

Related concepts

• open cover

• schedule

References

General

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

• Max Karoubi, Charles Weibel, On the covering type of a space, L’Enseignement Mathématique, 62 3/4 (2016) 457-474 [arXiv:1612.00532, doi:10.4171/LEM/62-3/4-4]

Applications of closed covers in Čech homology:

• Edwin E. Floyd, Closed coverings in Čech homology theory, Trans. Amer. Math. Soc. 84 (1957) 319-337 [doi:10.1090/S0002-9947-1957-0087100-2, 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 $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)