Compactness

Open covers, finite subcovers, Heine–Borel, and the extreme value theorem.

The idea

Compactness is the property that lets an infinite space be treated, in arguments, as though it were finite. For a finite set we can check a condition one point at a time, take the worst case, and conclude; for an infinite set the worst case may not exist at all. Compactness restores the finite argument — not by making the space small, but by guaranteeing that finitely many local pieces always suffice.

The local pieces are open sets. An open cover of $X$ is a collection of open sets whose union contains $X$, and a subcover is a subcollection that still covers.

Definition (Compact space).

A space $X$ is compact when every open cover of $X$ has a finite subcover.

The quantifiers matter: every cover must admit some finite subcover. Exhibiting one cover that has a finite subcover proves nothing, while exhibiting one cover with no finite subcover disproves compactness. The figure shows the open interval $(0, 1)$ together with a few of the sets $U_{n} = \left(\tfrac{1}{n}, 1\right)$; their union is all of $(0, 1)$, since every point of $(0, 1)$ exceeds $\tfrac{1}{n}$ once $n$ is large enough.

An open cover is exactly a collection of local information, one open set at a time, and passing to a finite subcover is what combines that information into a single global claim. That $[0, 1]$ is compact is the fact behind a continuous function on it being bounded and attaining its largest value.

Ways to work on it

• Walkthrough. The open-cover definition, Heine–Borel, and attained extrema.
• Practice. Decide whether a subset of the line is compact.
• Hardest. Extract a finite subcover from the open-cover definition by hand.

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