Covering

Definition

A covering of a set [ilmath]A[/ilmath] is a collection [ilmath]\mathcal{A} [/ilmath] where [math]A\subseteq\bigcup_{S\in\mathcal{A}}S[/math], that is as you'd expect, a collection of sets which contain [ilmath]A[/ilmath] in their union.

Alternative statement

Munkres seems to go a different route and only lets one cover entire spaces, not sets within it. However he shows that considering any set as a subspace of [ilmath]X[/ilmath] we can then cover it using (open, as it is brought up studying compactness) sets in the ambient space.

This is mentioned, discussed and proven on the compactness page.