Compactness

Let $(X, \tau)$ be a topological space.

Definition. An open cover of $X$ is a collection $\{U_i\}_{i \in I}$ of open sets with $\bigcup_{i \in I} U_i = X$. A subcover is a subcollection that still covers $X$. The space $X$ is compact if every open cover of $X$ has a finite subcover.

Proposition (Heine-Borel). A subset of $\mathbb{R}^n$ is compact (in the standard topology) if and only if it is closed and bounded.

Proposition (Extreme value theorem). If $X$ is compact and $f : X \to \mathbb{R}$ is continuous, then $f$ attains its supremum and infimum.

Proof sketch. $f(X)$ is compact in $\mathbb{R}$ (continuous image of a compact space is compact), hence closed and bounded by Heine-Borel. A closed bounded subset of $\mathbb{R}$ contains its supremum and infimum. $\square$

Proposition. A continuous map from a compact space to a Hausdorff space is a closed map. In particular, a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Theorem (Tychonoff). The product $\prod_{i \in I} X_i$ (with the product topology) is compact if and only if each $X_i$ is compact. (The forward direction requires the axiom of choice.)

Examples.

1. $[0,1]$. Compact (Heine-Borel, or directly by the nested interval property).
2. $\mathbb{R}$. Not compact: the cover $\{(-n, n)\}_{n \in \mathbb{N}}$ has no finite subcover.
3. Cantor set. Compact (closed and bounded subset of $\mathbb{R}$), despite being uncountable and nowhere dense.
4. Finite spaces. Every finite topological space is compact.