Covering

Definition

A covering of a set $A$ is a collection $\mathcal{A}$ where

$$A\subset\cup_{S\in\mathcal{A}}S$$ , that is as you'd expect, a collection of sets which contain $A$ 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 $X$ 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.