nLab
closed cover
Context
Topos Theory
topos theory

Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
Topology
topology (point-set topology )

see also algebraic topology , functional analysis and homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space (see also locale )

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

sphere , ball ,

circle , torus , annulus

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Basic homotopy 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 $X$ : $\cup_i U_i = X$ . 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
General
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

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 )

Revised on July 17, 2014 14:44:38
by

Urs Schreiber
(82.136.246.44)