Topology/Compactness

< Topology

The notion of Compactness appears in a wide variety of contexts. In particular, compactness is a "tameness property" that tells you that the objects you are dealing with are in some sense well-behaved.

Definition

Let be a topological space and let

A collection of open sets is said to be an Open Cover of if

is said to be Compact if and only if every open cover of has a finite subcover. More formally, is compact iff for every open cover of , there exists a finite subset of that is also an open cover of .

If the set itself is compact, we say that is a Compact Topological Space.

Compactness of topological spaces can also be expressed by one of the following equivalent characterisations:

Important Properties




Sources differ as to what exactly should be called the 'Heine-Borel Theorem'. It seems that Emile Borel proved the most relevant result, dealing with compact subsets of a Euclidean Space. However, we provide the simpler case, for reals.



Tychonoff's Theorem

The more general result on the compactness of product spaces is called Tychonoff's Theorem. Unlike the compactness of the product of two spaces, however, Tychonoff's Theorem requires Zorn's Lemma. (In fact, it is equivalent to Axiom of Choice.)

Theorem: Let , and let each be compact. Then the X is also compact.

Proof: The proof is in terms of nets. Recall the following facts:

Lemma 1 - A net in converges to if and only if each coordinate converges to .

Lemma 2 - A topological space is compact if and only if every net in has a convergent subnet.

Lemma 3 - Every net has a universal subnet.

Lemma 4 - A universal net in a compact space is convergent.

We now prove Tychonoff's theorem.

Let be a net in .

Using Lemma 3 we can find a universal subnet of .

It is easily seen that each coordinate net is a universal net in .

Using Lemma 4 we see that each coordinate net converges, because is compact.

Using Lemma 1 we see that the whole net converges in .

We conclude that every net in has a convergent subnet, so, by Lemma 2, must be compact.

Relative Compactness

Relative compactness is another property of interest.

Definition: A subset S of a topological space X is relative compact when the closure Cl(x) is compact.

Note that relative compactness does not carry over to topological subspaces. For example, the open interval (0,1) is relatively compact in R with the usual topology, but is not relatively compact in itself.

Local Compactness

The idea of local compactness is based on the idea of relative compactness.

If, in a topological space X, every element has a neighborhood that is relatively compact, then X is locally compact.

It can be shown that all compact sets are locally compact, but not conversely.

Exercises

  1. It is not true in general for a metric space that a closed and bounded set is compact. Take the following metric on a set X:
                           1   if x is not equal to y            
               d(x,y) =  
                           0   if x=y

a) Show that this is a metric

b) Which subspaces of X are compact

c) Show that if Y is a subspace of X and Y is compact, then Y is closed and bounded

d) Show that for any metric space, compact sets are always closed and bounded

e) Show that with this particular metric, closed and bounded sets need not be compact

This article is issued from Wikibooks. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.