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 $$X$$

Lemma for a set being compact

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 $$Y$$

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TODO: Proof

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