Definition:Subcover

Let $$S$$ be a set.

Let $$\mathcal{U}$$ be a cover for $$S$$.

A subcover of $$\mathcal{U}$$ for $$S$$ is a set $$\mathcal{V} \subseteq \mathcal{U}$$ such that $$\mathcal{V}$$ is also a cover for $$S$$.

Finite Subcover
A finite subcover of $$\mathcal{U}$$ for $$S$$ is a subcover $$\mathcal{U}$$ of which is finite.

Despite the obvious nature of its specification, it merits its own subdefinition because of its importance in the theory of compact topological spaces.