Definition:Cover of Set

Definition
Let $S$ be a set.

A cover (or covering) for $S$ is a set of sets $\mathcal C$ such that:
 * $\displaystyle S \subseteq \bigcup \mathcal C$.

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

Finite Cover
A cover $\mathcal C$ for $S$ is said to be finite iff $\mathcal C$ is a finite set.

Countable Cover
A cover $\mathcal C$ for $S$ is said to be countable iff $\mathcal C$ is a countable set.