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 $\mathcal U$ of which is finite.

Countable Subcover
A countable subcover of $\mathcal U$ for $S$ is a subcover $\mathcal U$ of which is countable.