Open cover

An open cover of a subset $A\subseteq X$ in a topological space $X$ is a cover $\{U_i\}_{i\in I}$ of $A$ such that each $U_i$ is an open set in $X$. Equivalently, $A\subseteq \bigcup_{i\in I}U_i$ and every $U_i$ is open in $X$.

Open covers are the basic input to the definition of compactness , and refinements of open covers are central in many “finiteness” arguments.

Examples:

• In $\mathbb{R}$ with the usual topology, the family $\{(n-1,n+1)\}_{n\in\mathbb{Z}}$ is an open cover of $\mathbb{R}$.
• In $\mathbb{R}$, the family $\{(-1/n,\,1+1/n)\}_{n\in\mathbb{N}}$ is an open cover of $[0,1]$.