Covering
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Definition
A covering of a set [ilmath]A[/ilmath] is a collection [ilmath]\mathcal{A} [/ilmath] where [math]A\subset\cup_{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.
