Compactness of Topological Space

Cover and Subcover

A cover of a set is collection of sets whose union contains : A subcover of a cover is a subset of whose elements still cover . A cover is open if all its elements are open.

e.g. is an open cover of ; is a subcover.

Compactness

A topological space is compact if every open cover of has a finite subcover. A subset of is compact if every open cover of by subsets of has a finite subcover. This is the same as being compact with the subspace topology.

e.g. is not compact, is an open cover with no finite subcover; is not compact, has no finite subcover.

Sequential Compactness

Let be a topological space and . We say that is sequentially compact if every sequence in has a subsequence converges to a point in .

Lemma

If is a topological space and then is compact in the if and only if is compact.

Theorem

Let be a compact topological space and a closed subspace of . Then is compact.

Proof Let be closed and let be an open covering of . Then is an open covering of . Since is compact, there exist such that It follows that , therefore is compact.

Proposition

Any compact subset of a Hausdorff space is closed.

Tikhonov Theorem

Def Product of Sets Let be a family of sets. We define to be the collection of all functions such that for each .

Def Projection

References and Other Resources

• Morphocular, The Concept So Much of Modern Math is Built On