Cover (topology)

In mathematics, and more particularly in set theory, a cover (or covering) of a set ${\displaystyle X}$ is a family of subsets of ${\displaystyle X}$ whose union is all of ${\displaystyle X}$. More formally, if ${\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace }$ is an indexed family of subsets ${\displaystyle U_{\alpha }\subset X}$ (indexed by the set ${\displaystyle A}$), then ${\displaystyle C}$ is a cover of ${\displaystyle X}$ if ${\displaystyle \bigcup _{\alpha \in A}U_{\alpha }\supseteq X}$. Thus the collection ${\displaystyle \lbrace U_{\alpha }:\alpha \in A\rbrace }$ is a cover of ${\displaystyle X}$ if each element of ${\displaystyle X}$ belongs to at least one of the subsets ${\displaystyle U_{\alpha }}$.

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.