relationship among different kinds of compactness

The goal of this article is to prove

Theorem 1.

If X is second countable and T1, or if X is a metric space, then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    X is compactPlanetmathPlanetmath;

  2. 2.
  3. 3.

We prove this using several subsidiary theorems, which prove the various implicationsMathworldPlanetmath in stronger settings.

Theorem 2.

A compact topological spaceMathworldPlanetmath T is limit point compact. (Here we make no assumptionsPlanetmathPlanetmath about the topology on T).

Proof. Choose an subset AT, and suppose A has no limit pointsMathworldPlanetmathPlanetmath. Then A contains its (vacuous set of) limit points and is therefore closed. But closed subsets of compact spaces are compact, so A is compact. Since A has no limit points, we may choose a neighborhood Ua of each aA such that Ua intersects A only in a. But this cover clearly has a finite subcover only if A is finite. So any set without limit points is finite, and thus any infinite setMathworldPlanetmath has a limit point. This concludes the proof.

Theorem 3.

If T is first countable, T1, and limit point compact, then T is sequentially compact.

Proof. Let xi be any sequenceMathworldPlanetmath of points in T, and assume that xi takes infinitely many values (otherwise it obviously has a convergentMathworldPlanetmathPlanetmath subsequence). Choose a limit point x for the sequence; we may assume wlog that xi is equal to x for only finitely many i (otherwise again the result holds trivially). So by ignoring a finite number of leading terms of the sequence, we may assume that xix for every i. Since T is first countable, choose a countable basis Bi at x; by replacing Bn with B1Bn, we may assume that Bi+1Bi for all i.

Now, choose n1 such that pn1B1. Inductively, assume we have chosen n1,,nk with pnkBk. Since T is T1, we may choose a neighborhood U of q that is disjoint from pn1,,pnk; choose pnk+1 to be any point in UBk+1. Then inductively the pni form a subsequence with pniBi, and clearly the pni converge to q. This concludes the proof.

Note that every metric space and every second countable T1 space is also first countable and T1.

Proposition 1.

Any sequentially compact metric space M is second countable.

Proof. It clearly suffices to show that M has a countableMathworldPlanetmath dense subset.

Claim first that for ϵ>0, the set of ϵ-balls in M has a finite subcover. Suppose this is false for some particular ϵ. Let p1M be any point, and construct inductively points pk with pkBϵ(p1)Bϵ(pk-1). Since M is sequentially compact, we may replace the pi by a convergent subsequence, which we also call pi, with pipM. But convergent sequences are Cauchy, so for n large enough, we have d(pn,pm)<ϵ, which contradicts the construction of the pi. This proves the claim.

Then for each positive integer n, let pn,n1,,pn,nkM be a finite setMathworldPlanetmath of points such that the 1n-balls around those points cover M. This set of points is countable, and is obviously dense in M. This concludes the proof.

Theorem 4.

If T is second countable or is a metric space, and sequentially compact, then T is compact.

Proof. Assume first that T is second countable. Choose any open cover of T; it has a countable subcover Ui. We use an argumentMathworldPlanetmath very similar to that used in the above propositionPlanetmathPlanetmath. Suppose no finite subset of the Ui covers T, and choose pkT\(U1Uk). Since T is sequentially compact, the pk have a convergent subsequence pnk converging to pT. But pUn for some n; since the pnk converge to p, all pnkUn for k large enough. But this is a contradictionMathworldPlanetmathPlanetmath to the construction of the pk, so that a finite subset of the Ui cover T and T is compact.

Since any sequentially compact metric space is second countable by the above proposition, we are done.

The main theorem follows trivially from the above. Note that we have in fact proven the following set of implications:

  • Compact limit point compact for general topological spaces;

  • Limit point compact sequentially compact for first countable T1 spaces;

  • Sequentially compact compact for second countable or metrizable spaces.

Title relationship among different kinds of compactness
Canonical name RelationshipAmongDifferentKindsOfCompactness
Date of creation 2013-03-22 18:00:57
Last modified on 2013-03-22 18:00:57
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 40A05
Classification msc 54D30