Compactness in Metric Spaces

 The idea of compactness comes from the fact that these are the nice set next to the finite sets. i.e. It shares many properties of finite sets in a topological space. The first thing common between finite and compact sets is, these are closed and bounded. Every sequence has a limit point. A continuous image of a compact set is compact and many more. Compactness has a very close relation to completeness.

Let X be a topological space then the definition of compactness is defined in the following way-:

X is said to be compact if every open covering of X has a finite sub-covering.

Heine-Borel Theorem -: Heine-Borel make our work easy for Euclidean spaces by generalizing that, ‘K is a compact subset of R if and only if it is closed and bounded.’

But above theorem doesn’t work well for all metric spaces, for example -: Take N with discrete metric. What is missing here? Completeness? No! Boundedness? No! Closedness? No!

Then what?

The thing that is missing is totally boundedness.

The idea of totally boundedness arises from the fact that for a compact set the trivial covering should also have a finite subcover. So, one can define totally boundedness in terms of covering-: every trivial cover has a finite subcover.

N with discrete metric is not compact, because if we take the cover consisting ball of radius ϵ > 0 around every point, then it will not have a finite subcover if the radius of balls is less than 1.

Note-: Totally boundedness is necessary for compactness in a metric space, but not sufficient.

Counterexample -:  Open interval (0,1) is totally bounded but it’s not compact. More generally, a subset of a compact set is totally bounded.

Bounded does not imply totally bounded.

Now we will be focusing on some of the important characterizations of compact sets. Let X be a metric space then, TFAE-:

1.     X is compact i.e. every open cover of X has a finite subcover.

2.     X is complete and totally bounded.

3.     Every infinite subset of X has a cluster point.

4.     Every sequence in X has a convergent subsequence. (BWP)

(1 2) Totally bounded is a trivial consequence of being. For completeness property, we will construct a sequence of open cover from a subsequence of a given Cauchy sequence whose finite subset will not cover X, because of compactness union of this sequence will not cover X and the element which is not in the union is the limit of the subsequence.

(2 3) The main thing is to construct a Cauchy sequence. Every 1/n net has a finite subcover so at least one of these has an infinite element of our set. Construct a Cauchy sequence from these sets.

(3 4) Trivial.

(4 1) X is bounded.  

 

Bolzano Weierstrass Theorem-: It says that every infinite bounded subset of R has a cluster point.

Bolzano Weierstrass Property-: A set S in a metric space has the Bolzano-Weierstrass Property if every sequence in S has a convergent subsequence — i.e., has a subsequence that converges to a point in S.