Concept

Cover (topology)

Summary
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X (indexed by the set A), then C is a cover of X if \bigcup_{\alpha \in A}U_{\alpha} = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_{\alpha}. A subcover of a cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is a col
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