We started off reviewing the definition of compactness and the properties we proved of it. Then we proved that in \(\mathbb{R}\), a set is compact if and only if it is closed and bounded. As an application of this, we proved that any continuous function \(f : [0, 1] \rightarrow \mathbb{R}\) has a maximum, i.e. a point \(a \in [0, 1]\) such that \(f(x) \le f(a)\) for every \(x \in [0, 1]\). Students pointed out that this result still holds as long as the domain is any compact set, and the co-domain is any linearly-ordered set with the order topology (we aren’t covering order topologies in this course).

We spent the second half proving a subtle but useful fact which says that compactness can be verified by only considering open covers coming from some basis. More specifically, let \(X\) be a topological space with basis \(\mathcal{B}\). Then a set \(K \subseteq X\) is compact if an only if for every open cover \(\mathcal{O}\) consisting of open sets in \(\mathcal{B}\), there exists a finite subcover. The forward direction is immediate, but the backwards direction required careful argument.