Definition:Cover of Set

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 $$\mathcal{U}$$.