Cover

Definition
A cover of a set (usually a topological space) is a family of subsets of the set whose union is the whole set.

Properties of covers

 * Open cover is a cover comprising open sets
 * Closed cover is a cover comprising closed sets

In general, given any property of collections of subsets, we can evaluate that property for covers.

Related notions
Related to covers, we have notions like subcover (a subset of the family of subsets of the cover, that again forms a cover), refinement (another family of subsets such that each subset of the new family is contained in a subset of the old family), shrinking and expansion.