nLab
open cover

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Definition

An open cover of a topological space X is a collection {U i⊂X} of open subsets of X whose union equals X: ∪ iU i=X.

Properties

When denoting by U i↪X the inclusion morphisms in the category Top, each open cover constitutes a covering family {U i→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 manifolds.

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 April 29, 2012 13:05:19 by Urs Schreiber (82.113.119.38)
Edit | Back in time (5 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Diff, hypercover, site, smooth infinity-groupoid, CartSp, cover, simplicial complex, finite object, coverage, Grothendieck pretopology, model structure on simplicial presheaves, van Kampen theorem, internal hom, category of sheaves, structured (infinity,1)-topos, topological stack, extended natural number, factorization algebra, local homeomorphism, gerbe, numerable open cover, paracompact topological space, twisted bundle, refinement, topological K-theory, totally bounded space, foliation, Euclidean-topological infinity-groupoid, concrete site, open subspace, partition of unity, topological localization, Warsaw circle, locale of real numbers, determinant line bundle, smooth manifold, universal Chern-Simons circle 3-bundle, BDR 2-vector bundle, good open cover, locally finite cover, descent for L-infinity algebras, Cahiers topos, formal topology, function algebras on infinity-stacks, split hypercover, infinity-Chern-Weil theory introduction, connection on a 2-bundle, dimension, uniform locale, cohesive topos, TopMfd, cohesive site, locally connected site, covering dimension, morphism of finite type, classical triangulation, smooth infinity-groupoid -- structures, connected site, dense sub-site, small-generated site, uniform cover, locally contractible topological infinity-groupoid, fermionic path integral, universal principal bundle, Mayer-Vietoris sequence, schedule, closed cover, twisted smooth cohomology in string theory, pullback of a differential form, geometry of physics