Compact Sets (Heine–Borel)

Compact means closed and bounded — every open cover has a finite subcover.

The idea

Heine–Borel. A subset $K \subseteq \mathbb{R}^{n}$ is compact if and only if it is closed and bounded.

Compactness itself is defined by covers. A collection of open sets whose union contains $K$ is an open cover of $K$, and $K$ is compact when every open cover of $K$ contains finitely many sets that still cover $K$. The strength of the condition lies in its first quantifier: any set sits inside a single open set, so demanding that some cover be finite would demand nothing. Compactness demands that a finite subcover can be extracted from every cover, however that cover was chosen. The figure shows an open cover of the interval $(0, 1)$: a few of the sets $U_{n} = \left(\tfrac{1}{n}, 1\right)$, whose left ends creep toward $0$ while their union is all of $(0, 1)$.

This is the property analysis uses. An argument that works only locally, in a small neighborhood of each point, extends through finitely many neighborhoods to a statement about all of $K$ — and finitely many bounds have a largest, finitely many tolerances a smallest, while infinitely many need have neither.

Checking every open cover directly is impractical, and that is the value of Heine–Borel: within $\mathbb{R}^{n}$ it replaces the covering condition by two conditions we can check by inspection. Each seals off one way a point can escape a set. Bounded rules out points running away to infinity; closed rules out points accumulating at a boundary point the set does not contain.

Ways to work on it

• Walkthrough. Heine–Borel, examples of compact sets, and the open-cover definition.
• Practice. Decide whether a given subset of the line is compact.
• Hardest. Build an open cover of a half-open interval with no finite subcover.

Not sure where to start? Take the ten-question placement test.