Chapter 4 (compact topological spaces)

Open cover definition

An open cover is finite if the set indexing the family is finite.

Subcover definition

Compact definition

A topological space X is compact if every open cover for X has a finite subcover.

A subset K of X is compact if it is compact in the subspace topology.

When is a subset compact?

A subset K of X is compact if and only if every open cover for K has a finite subcover.

Finite subset compactness

Any finite subset of X is compact, for any topological space.

Supremum definition

The least upper bound of a set

Closed intervals theorem

Closed intervals are compact in the standard topology.

Continuous image theorem

This theorem can be used to prove compactness, i.e. if we know a set can be written as the continuous image of a compact set, then we know the set is compact.

Closed subset - compact space proposition

A closed subset of a compact space is compact

Compact subset - Hausdorff proposition

A compact subset of a Hausdorff space is closed

When is a subset compact? (relating to closed-ness)

Finite union corollary

If A, B are compact subsets of X, then A U B is a compact subset of X.

Arbitrary intersection corollary

In a Hausdorff space, arbitrary intersections of compact subsets are compact.

Slice and tube definition

Inclusion map lemma

Tube lemma

Tychonoff theorem

It follows by induction that any finite product of compact topological spaces is compact.

Closed interval - compactness lemma

Bounded definition

Note that boundedness is a property of metric spaces and not of topological spaces.

However if a subset is compact and a topology arises from a metric, then the subset is bounded in all distances in the topology.

Closed - bounded proposition

Heine-Borel theorem

Supremum-adherent point lemma

Maximum - minimum theorem

Homeomorphism definition

Two spaces which are homeomorphic are topologically the same, up to a relabelling of the points.

Homeomorphism proposition

Preimage - continuity lemma

Hausdorff - homeomorphism theorem

What does it mean for homeomorphisms to be equivalence relations?

Topological invariant definition

In simpler terms a topological invariant is a property of a topological space X, such that given any other space Y, if Y is homeomorphic to X, then Y has it too.

We can use this to prove that two spaces are not homeomorphic, but not the other way around.

What are the three topological invariants?