Applying compactness

This is a survey article about applying the concept/definition/theorem: compactness

This article is about how one can use the point set-topological fact that a given topological space (possibly, with a lot of additional strucure) is compact.

The open cover formulation

Let us recall the open cover formulation of compactness:

For a compact space, every open cover has a finite subcover

This is one of the most useful formulations of compactness, and one way to think of it is as a black box: feed in any open cover to it, and out comes a finite subcover. As such, we have no control over the subcover apart from the fact that it is finite and is a subcollection of the original open cover, so how well we use the hypothesis of compactness depends on how well we choose our initial open cover.

The typical choice for the initial open cover is a point-indexed open cover; for every point, we choose a suitable, sufficiently small, open set around it satisfying particularly nice properties. Point-indexed open covers help us quantitatively describe exactly what properties we need around each point, and they also ensure that we do get a cover (because every point is in the open set indexed by it).

Use in proving separation results

Examples of separation results that are proved using compactness are:

The key idea is both these proofs is as follows: we start with a compact subset $A$ of $X$ , which we want to separate from some point $x$ outside. For every point $y \in A$ , we use Hausdorffness to get disjoint open sets around $x$ and $y$ . The disjoint open set about $y$ is the open set for $y$ , and we use this construction to get a point-indexed open cover. Compactness then allows us to pass to a finite subcover; the advantage of doing this is that we then have only finitely many open subsets to intersect on the other side.

Use in proving results about spaces with local models

Another powerful application of the open cover formulation of compactness is to the study of spaces with local models, for instance, manifolds, differential manifolds, real-analytic manifolds, fiber bundles, and the like. In each of these situations, we have a local model condition. For a manifold, the local model is Euclidean space (topologically); for a differential manifold, it is Euclidean space (differentially), for a fiber bundle, the local model is a projection map from a product space.

To use compactness of a space with such a local structure, we typically proceed to construct our point-indexed open cover as follows:

For each point, find a sufficiently small open neighbourhood of the point where the local model is applicable
Refine this to an even smaller neighbourhood; often we pick a neighbourhood whose closure lies completely inside a neighbourhood where the local model is applicable (this sometimes requires an assumption of the space being Hausdorff, to ensure that compact subsets are closed)
Now try to find an even smaller neighbourhood within this where the global property that we seek, is true. Be sure to take as small a neighbourhood as possible before applying the finite subcover machine

Once we have a point-indexed open cover, we obtain a finite subcover. We then try to use the finite subcover in one of many possible ways:

A finite subcover allows us to talk of minima and maxima instead of infima and suprema; this might enable us to make a global choice by taking a minimum of whatever works for each member of the subcover. An example is the tubular neighbourhood theorem in differential geometry
A finite subcover allows us to embed in a product, using a partition of unity; this is the idea behind the Whitney embedding theorem

Multiple iterations of the open cover argument; or further use of compactness

In some situations, we need to use compactness again after applying the open cover formulation once; we may need to do another open cover formulation, or we may need to use one of the many other implications of compactness. An example where we need to use the open cover argument twice is to show that compact Hausdorff implies normal: the first iteration shows regularity using Hausdorffness; the second iteration shows normality using regularity.

The finite intersection property formulation

This formulation is very readily seen to be equivalent to the open cover formulation, but is at time a more convenient way of thinking. A collection of subsets of a space is said to have the finite intersection property if any finite intersection is nonempty. Then, in a compact space, any collection of closed subsets with the finite intersection property, has nonempty intersection.

This formulation is sometimes more natural because closed subsets arise as the zero sets of collections of functions. It can thus be used to show that if a collection of functions does not have a common zero, then there is a finite subcollection that does not have a common zero.

Compact spaces and continuous real-valued functions

Consider the ring of continuous real-valued functions on a space. For every point in the space, the ideal of functions that vanish at that point, is a maximal ideal. It turns out that when the space is compact, then the maximal ideals are precisely those that come from points.

Essentially, this boils down to showing that if we have a proper ideal in the ring of continuous functions, there exists a point at which all the functions in the ideal vanish. The argument to prove this starts by observing that for every function, the closed subsets at which they vanish have the finite intersection property. In other words, we show that it cannot happen that finitely many functions have no common vanishing point (i.e. if that happens, the ideal is no longer proper). We then use copmactness to argue that the intersection of all the closed subsets is nonempty.

Compact Hausdorff spaces are Baire

Further information: Compact Hausdorff implies Baire

The idea here is to construct a descending chain of closed subsets in such a fashion that any point in their intersection proves the result we want. The existence of a point in the intersection is guaranteed by compactness.

The sequential compactness and limit point formulation

Further information: Sequentially compact space, limit point-compact space

Any compact space is sequentially compact: every sequence has a convergent subsequence. For metrizable spaces, the converse is also true, and so, in a number of applications, we do not lose anything by viewing compactness as sequential compactness.

Any compact space is limit point-compact: every infinite set has a limit point. Again, for metrizable spaces, the converse is also true.

The Lebesgue number lemma

This is a very useful application of compactness to metric spaces, and is often used along with other methods, to establish properties for compact metric spaces.