Covers

Definition: Cover

Let $X$ be a set.

A collection $\mathcal{C}$ of subsets of $X$ is a cover of $X$ if $X$ is a subset of the union of $\mathcal{C}$.

$$X\subseteq \bigcup \mathcal{C}$$

Definition: Subcover

Let $X$ be a set and let $\mathcal{C}$ be a cover of $X$.

A subcover of $\mathcal{C}$ is subcollection of $\mathcal{C}$ which is still a cover of $X$.