nLab
open cover

Contents

Definition

An open cover of a topological space XX is a collection {U iX}\{U_i \subset X\} of open subsets of XX whose union equals XX: iU i=X\cup_i U_i = X.

Properties

When denoting by U iXU_i \hookrightarrow X the inclusion morphisms in the category Top, each open cover constitutes a covering family {U iX}\{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 manifolds.

If an open cover has the property that all the U iU_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)