Cover (Topology)

A cover of a set $X$ is a family of subsets of $X$ whose union is all of $X$. More formally, if $C=\left\{U_{\alpha }:\alpha \in A\right\}$ is an indexed family of subsets $U_{\alpha }\in X$ (indexed by the set $A$), then $C$ is a cover of $X$ if $X\subset {\bigcup }_{\alpha \in A}{U}_{\alpha }$