Cover - Knowlpedia

A cover of a subset $A\subseteq X$ is a family $\{U_i\}_{i\in I}$ of subsets of $X$ such that

A \subseteq \bigcup_{i\in I} U_i.

Covers are often presented as an indexed family of sets, and the condition above says that the union of the family contains $A$ . In topology, one frequently restricts to an open cover.

Examples

In $\mathbb{R}$ , the family $\{(n-1,n+1)\}_{n\in\mathbb{Z}}$ covers $\mathbb{R}$ .
For $A=[0,1]\subseteq \mathbb{R}$ , the sets $U_1=(-1,1/2)$ and $U_2=(0,2)$ form a cover of $A$ .