Definition:Cover of Set

Definition
Let $S$ be a set.

Then a cover for $S$ (or covering) is a set $\mathcal U$ of sets such that:
 * $\displaystyle S \subseteq \bigcup_{U \in \mathcal U} U$

We say that $S$ is covered by $\mathcal U$.

Finite Cover
A cover $\mathcal U$ is finite iff there are only finitely many sets in $\mathcal U$.

Countable Cover
A cover $\mathcal U$ is countable iff there are only countably many sets (either finitely or not) in $\mathcal U$.