nLab
open cover
Context
Topology
topology (point-set topology , point-free topology )

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

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

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

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

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

Analysis Theorems

topological homotopy theory

Contents
Definition
An open cover of a topological space $X$ is a collection $\{U_i \subset X\}$ of open subset s of $X$ whose union equals $X$ : $\cup_i U_i = X$ .

Properties
When denoting by $U_i \hookrightarrow X$ the inclusion morphisms in the category Top , each open cover constitutes a covering family $\{U_i \to X\}$ in the sense of sheaf and topos theory which is a standard coverage on Top .

Analogous statements hold for categories of topological spaces with extra structure , such as the category Diff of smooth manifold s.

If an open cover has the property that all the $U_i$ and all of their finite nonempty intersections are contractible , then one speaks of a good open cover .

Revised on June 3, 2017 11:15:30
by

Todd Trimble
(67.81.95.215)