S. D. V. Stephens

notes / Math 55b / chapter 2: Connectedness and Compactness / From my notebook

N12 Compactness

Compactness is a finiteness/boundedness property of nice topological spaces, such as closed intervals and closed and bounded subsets of .

E.g. any continuous map with compact achieves its min and max.

Definition N12.1. An open cover of a topological space is a collection of open subsets such that .

Definition N12.2. is compact if every open cover of admits a finite subcover, i.e. there exist such that .

Example N12.3. is not compact with the usual topology: , but there is no finite subcover. The integer lies only in , so we cannot discard any of the intervals and still have a cover.

Example N12.4. Finite sets are compact. Also is compact: let be an open cover of . There is such that , and since is open it also contains for all large enough .

Proposition N12.5 (Claim). is compact in with the usual topology.

Proof. Take an open cover of . Let because . We want to show . Let .

First we show , i.e. that has a finite subcover. There is with . Since is open, there is such that . But , so there is with . So there are such that , and then .

The interval [0,1] with a point a, an open set U_{i_0} around a containing (a - epsilon, a + epsilon), a point x of A just above a - epsilon, and the interval [0,x] covered by finitely many U_i.

Now assume . Since , there are such that . This union is open and contains , so it contains for some . Take with ; then , so , which contradicts . So , and there is a finite subcover of . ∎