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.