Open cover - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $X$ .

Equivalent characterizations

Equivalently, $A\subseteq \bigcup_{i\in I}U_i$ and every $U_i$ is open in $X$ .

Remarks

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 $\{(-1/n,\,1+1/n)\}_{n\in\mathbb{N},\ n\ge 1}$ is an open cover of $[0,1]$ .