# Definition:Subcover

## Definition

Let $S$ be a set.

Let $\mathcal U$ be a cover for $S$.

A subcover of $\mathcal U$ for $S$ is a set $\mathcal V \subseteq \mathcal U$ such that $\mathcal V$ is also a cover for $S$.

### Finite Subcover

A finite subcover of $\mathcal U$ for $S$ is a subcover $\mathcal V \subseteq \mathcal U$ which is finite.

### Countable Subcover

A countable subcover of $\mathcal U$ for $S$ is a subcover $\mathcal V \subseteq \mathcal U$ which is countable.