Compactness

Definition

A topological space is compact if every open cover (often denoted

$\mathcal{A}$ ) of

$X$ contains a finite sub-collection that also covers

$X$

Take a set

$Y\subset X$ in a topological space

$(X,\mathcal{J})$ .

To say

$Y$ is compact is for

$Y$ to be compact when considered as a subspace of

$(X,\mathcal{J})$

That is to say that

$Y$ is compact if and only if every covering of

$Y$ by sets open in

$X$ contains a finite subcovering covering

$Y$

TODO: Proof