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 $$Y$$

Proof

$$\implies$$

Suppose that the space

$$(Y,\mathcal{J}_\text{subspace})$$ is compact and that $$\mathcal{A}=\{A_\alpha\}_{\alpha\in I}$$ where each $$A_\alpha\in\mathcal{J}$$ (that is each set is open in $$X$$ ).

Then the collection

$$\{A_\alpha\cap Y|\alpha\in I\}$$ is a covering of $$Y$$ by sets open in $$Y$$ (by definition of being a subspace)

By hypothesis

$$Y$$ is compact, hence a finite sub-collection $$\{A_{\alpha_i}\cap Y\}^n_{i=1}$$ covers $$Y$$ (as to be compact every open cover must have a finite subcover)

Then

$$\{A_{\alpha_i}\}^n_{i=1}$$ is a sub-collection of $$\mathcal{A}$$ that covers $$Y$$ .

Details

As The intersection of sets is a subset of each set and

$$\cup^n_{i=1}(A_{\alpha_i}\cap Y)=Y$$ we see
$$x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies\exists k\in\mathbb{N}\text{ with }1\le k\le n:x\in A_{\alpha_k}\cap Y$$ $$\implies x\in A_{\alpha_k}\implies x\in\cup^n_{i=1}A_{\alpha_i}$$
The important part being $$x\in\cup^n_{i=1}(A_{\alpha_i}\cap Y)\implies x\in\cup^n_{i=1}A_{\alpha_i}$$
then by the implies and subset relation we have $$Y=\cup^n_{i=1}(A_{\alpha_i}\cap Y)\subset\cup^n_{i=1}A_{\alpha_i}$$ and conclude $$Y\subset\cup^n_{i=1}A_{\alpha_i}$$

Lastly, as

$$\mathcal{A}$$ was a covering $$\cup_{\alpha\in I}A_\alpha=Y$$ .

It is clear that

$$x\in\cup^n_{i=1}A_{\alpha_i}\implies x\in\cup_{\alpha\in I}A_\alpha$$ so again implies and subset relation we have:
$$\cup^n_{i=1}A_{\alpha_i}\subset\cup_{\alpha\in I}A_\alpha=Y$$ thus concluding $$\cup^n_{i=1}A_{\alpha_i}\subset Y$$

Combining

$$Y\subset\cup^n_{i=1}A_{\alpha_i}$$ and $$\cup^n_{i=1}A_{\alpha_i}\subset Y$$ we see $$\cup^n_{i=1}A_{\alpha_i}=Y$$

Thus

$$\{A_{\alpha_i}\}^n_{i=1}$$ is a finite covering of $$Y$$ consisting of open sets from $$X$$

$$\impliedby$$

Suppose that every covering of

$$Y$$ by sets open in $$X$$ contains a finite subcollection covering $$Y$$ . We need to show $$Y$$ is compact.

Suppose we have a covering,

$$\mathcal{A}'=\{A'_\alpha\}_{\alpha\in I}$$ of $$Y$$ by sets open in $$Y$$

For each

$$\alpha$$ choose an open set $$A_\alpha$$ open in $$X$$ such that: $$A'_\alpha=A_\alpha\cap Y$$

Then the collection

$$\mathcal{A}=\{A_\alpha\}_{\alpha\in I}$$ covers $$Y$$

By hypothesis we have a finite sub-collection of things open in

$$X$$ that cover $$Y$$

Thus the corresponding finite subcollection of

$$\mathcal{A}'$$ covers $$Y$$