Open cover

Definition

Let $X$ be a topological space and let $A \subseteq X$. An open cover of $A$ is a family $(U_i)_{i \in I}$ of open subsets of $X$ with $A \subseteq \bigcup_{i \in I} U_i$. A subcover is a subfamily $(U_j)_{j \in J}$, with $J \subseteq I$, that is still an open cover of $A$. A subcover is finite when $J$ is a finite set.

Remarks

The intervals $(-n, n)$ for $n \ge 1$ form an open cover of $\mathbb{R}$. It has no finite subcover. The intervals $(1/n, 1)$ for $n \ge 2$ form an open cover of $(0, 1)$ with no finite subcover. A set $K$ is compact when each open cover of $K$ has a finite subcover.

Explanations

L1 human

An open cover is a collection of open sets that together cover the whole set. You can think of them as patches, and a finite subcover uses only finitely many patches to cover the set.