Cover

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

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 $A$.