Compact space

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In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).

A more modern approach is to call a topological space compact if all its open covers have a finite subcover. The Heine-Borel theorem affirms that this coincides with "closed and bounded" for subsets of Euclidean space.

Note: Some authors such as Bourbaki use the term "quasi-compact" instead and reserve the name "compact" for topological spaces that are Hausdorff and compact.

History and motivation

The term compact was introduced by Fréchet in 1906.

It has been recognized for a long time that a property like compactness was needed to prove many useful results. At one time, when primarily metric spaces were studied, compact was taken to mean sequentially compact (every sequence has a convergent subsequence). The definition based on open coverings has surpassed it by allowing many useful results that could be proven about metric spaces using the old definition to be proven in general.

One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets. In other words, there are many results which are easy to show for finite sets, the proofs of which carry over with minimal change to compact spaces. It is often said that "compactness is the next best thing to finiteness". Here is an example:

  • Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. Then we can separate x and A by neighbourhoods: for each a in A, let U(x) and V(a) be disjoint neighbourhoods containing x and a, respectively. Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A.

Note that if A is infinite, the proof fails, because the intersection of arbitrarily many neighbourhoods of x might not be a neighbourhood of x. The proof can be "rescued", however, if A is compact: we simply take a finite subcover of the cover {V(a)} of A. In this way, we see that in a Hausdorff space, any point can be separated by neighbourhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighbourhoods -- note that this is precisely what we get if we replace "point" (i.e. singleton set) with "compact set" in the Hausdorff separation axiom. Many of the arguments and results involving compact spaces follow such a pattern.

Definitions

Compactness of subsets of Rn

For any subset of Euclidean space Rn, the following four conditions are equivalent:

  • Every open cover has a finite subcover. This is the definition most commonly used.
  • Every sequence in the set has a convergent subsequence, the limit point of which belongs to the set.
  • Every infinite subset of the set has an accumulation point in the set.
  • The set is closed and bounded. This is the condition that is easiest to verify, for example a closed interval or closed n-ball.

In other spaces, these conditions may or may not be equivalent, depending on the properties of the space.

Compactness of topological spaces

The "finite subcover" property from the previous paragraph is more abstract than the "closed and bounded" one, but it has the distinct advantage that it can be given using the subspace topology on a subset of Rn, eliminating the need of using a metric or an ambient space. Thus, compactness is a topological property. In a sense, the closed unit interval [0,1] is intrinsically compact, regardless of how it is embedded in R or Rn.

The general definition goes as follows. A topological space is called compact iff all its open covers have a finite subcover. Formally, this means that

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Some authors require that a compact space also be Hausdorff, and the non-Hausdorff version is then called quasicompact.

Examples of compact spaces

Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions):

  • A continuous image of a compact space is compact.
  • A closed subset of a compact space is compact.
  • A compact subset of a Hausdorff space is closed.
  • A nonempty compact subset of the real numbers has a greatest element and a least element.
  • A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine-Borel theorem)
  • A metric space (or uniform space) is compact if and only if it is complete and totally bounded.
  • The product of any collection of compact spaces is compact. (Tychonoff's theorem -- this is equivalent to the axiom of choice)
  • A compact Hausdorff space is normal.
  • Every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
  • A metric space is compact if and only if every sequence in the space has a convergent subsequence.
  • A topological space is compact if and only if every net on the space has a convergent subnet.
  • A topological space is compact if and only if every filter on the space has a convergent refinement.
  • A topological space is compact if and only if every ultrafilter on the space is convergent.
  • A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.
  • Every topological space X is a dense subspace of a compact space which has at most one point more than X. (Alexandroff one-point compactification)
  • A metric space X is compact if and only if every metric space homeomorphic to X is complete.
  • If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (Lebesgue's number lemma)
  • If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact. (Alexander's Sub-base Theorem)
  • Two compact Hausdorff spaces X1 and X2 are homeomorphic if and only if their rings of continuous real-valued functions C(X1) and C(X2) are isomorphic.

Other forms of compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.

  • Sequentially compact: Every sequence has a convergent subsequence.
  • Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
  • Pseudocompact: Every real-valued continuous function on the space is bounded.
  • Weakly countably compact (or limit point compact): Every infinite subset has an accumulation point.

While all these conditions are equivalent for metric spaces, in general we have the following implications:

  • Compact spaces are countably compact.
  • Sequentially compact spaces are countably compact.
  • Countably compact spaces are pseudocompact and weakly countably compact.

Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology. Not every compact space is sequentially compact; an example is the infinite product space 2 [0, 1] with the product topology.

A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces. For complete metric spaces this is equivalent to compactness. See relatively compact for the topological version.

Another related notion that is usually strictly weaker than compactness is local compactness.

References

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