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 $$S \subseteq \bigcup_{U \in \mathcal U} U$$.

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 $$\mathcal U$$.