Definition:Cover

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 $\mathcal U$.

Also see

• Results about covers can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$