nLab closed cover

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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.

Often it is also required that every point xXx \in X is in the interior of one of the U iU_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 pp-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.