• \(X\) is open iff \(\overline{X}\) is closed.
• (but sets can be both open and closed)

In a general topology we just define which sets are open, and they must obey axioms. We say \(\tau\subseteq \mathcal{P}(X)\) is a topology on space \(X\) if it satisfies the following properties:

1. \(\varnothing, X \subset \tau\).
2. Any union of elements of \(\tau\) is an element of \(\tau\).
3. Any intersection of finitely many elements of \(\tau\) is an element of \(\tau\).

• complete space: limits exist.
• \(Y\) is compact if every open cover has a finite sub-cover

Often we work in metric space. In a metric space:

• \(Y\) is an open set iff for every point \(y\in Y\) there is a ball (aka neighborhood) of \(y\) contained in \(Y\).
• \(Y\) is a closed set iff for every convergent sequence \(y_1,y_2,\ldots \in Y\) the limit is contained in \(Y\).
• sequentially compact: every sequence has a convergent subsequence.

Metric spaces are compact iff they are sequential compact.

Some usefull things to know: A closed subset \(Y\) of a compact set \(X\) is compact. Proof is nice. Take an open cover of \(Y\). Extend to an open cover of \(X\) by adding \(Y\setminus X\) which is an open set because \(Y\) is a closed set. Then find a finite subcover for \(X\). Then toss \(Y\setminus X\) from that subcover if it was in it. This is now a finite subcover for \(Y\).