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Compact space

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"Compact" redirects here. For other uses, see Compact (disambiguation).

In mathematics, more specifically general topology and metric topology, an abstract mathematical space is said to be compact if intuitively, it is not too "large." Thus, while disks and spheres are compact, infinite lines and planes are not. Compactness generalizes many important properties of closed and bounded intervals in the real line; that is, intervals of the form [a,b] for real numbers a and b. For instance, any continuous function defined on a compact space into an ordered set (with the order topology) such as the real line is bounded. Thus, what is known as the extreme value theorem in calculus generalizes to compact spaces. In this fashion, one can prove many important theorems in the class of compact spaces, that do not hold in the context of non-compact ones.

During the earlier part of the 20th century, mathematicians attempted to generalize the notion of compactness to spaces outside Euclidean space. Since Euclidean space is a metric space and possesses other important properties, various notions of compactness such as sequential compactness and limit point compactness coincide for compact subspaces of Euclidean space. Therefore, when generalizing the notion of compactness, one must choose the single notion of compactness satisfied by subspaces of Euclidean space, that allows nice theorems to be proved, and this is somewhat difficult.

Formally, a topological space is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact.

The Heine–Borel theorem shows that this definition is equivalent to "closed and bounded" for subsets of Euclidean space. So 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 single compact set is sometimes referred to as a compactum; following the Latin second declension (neuter), the corresponding plural form is compacta.

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[edit] History and motivation

The identity of bounded closed subsets of real numbers and sets whose open covers have finite subcovers was discovered and proved in the late 19th century. See Heine–Borel theorem.

The term compact was introduced by Fréchet in 1906.

It has long been recognized that a property like compactness is necessary to prove many useful theorems. It used to be that "compact" meant "sequentially compact" (every sequence has a convergent subsequence). This was when primarily metric spaces were studied. The "covering compact" definition has become more prominent because it allows us to consider general topological spaces, and many of the old results about metric spaces can be generalized to this setting. This generalization is particularly useful in the study of function spaces, many of which are not metric spaces.

One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. 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) : a in A} of A, then intersect the corresponding finitely many U(x). 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.

[edit] Definitions

[edit] Compactness of topological spaces

A topological space X is defined as compact if all its open covers have a finite subcover. Formally, this means that

for every arbitrary collection \{U_\alpha\}_{\alpha\in A} of open subsets of X such that \bigcup_{\alpha\in A} U_\alpha \supseteq X, there is a finite subset J\subset A such that \bigcup_{i\in J} U_i \supseteq X.

An often used equivalent definition is given in terms of the finite intersection property: if any collection of closed sets satisfying the finite intersection property has nonempty intersection, then the space is compact[1]. This definition is dual to the usual one stated in terms of open sets.

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

[edit] 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 topological definition.
  • 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 at least one 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.

Note that while compactness is a property of the set itself (with its topology), closedness is relative to a space it is in; above "closed" is used in the sense of closed in Rn. A set which is closed in e.g. Qn is typically not closed in Rn, hence not compact.


[edit] Examples of compact spaces

  • The closed unit interval [0, 1] is compact. This follows from the Heine–Borel theorem; the proof of which is about as hard as proving directly that [0, 1] is compact. The open interval (0, 1) is not compact: the open cover ( 1/n, 1−1/n ) for n = 3, 4, …  does not have a finite subcover. Similarly, the set of rational numbers in the closed interval [0, 1] is not compact: the sets of rational numbers in the intervals \left[0,\frac{1}{\pi}-\frac{1}{n}\right] and \left[\frac{1}{\pi}+\frac{1}{n},1\right] cover all the rationals in [0, 1] for n = 4, 5, …  but this cover does not have a finite subcover. (Note that the sets are open in the subspace topology even though they are not open as subsets of R.)
  • The set R of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals (n − 1, n + 1), where n takes all integer values in Z, cover R but there is no finite subcover.
  • For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
  • Consider the set K of all functions f: \mathbb{R} \rightarrow [0,1] from the real number line to the closed unit interval, and define a topology on K so that a sequence {fn} in K converges towards f\in K if and only if {fn(x)} converges towards f(x) for all x\in\mathbb{R}. There is only one such topology; it is called the topology of pointwise convergence. Then K is a compact topological space, again a consequence of Tychonoff's theorem.
  • Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point compactification of R is homeomorphic to the circle S1; the one-point compactification of R2 is homeomorphic to the sphere S2. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
  • In the cocountable topology on R (or any uncountable set for that matter), no infinite set is compact.

[edit] Theorems

Some theorems related to compactness (see the glossary of topology for the definitions):

Characterizations of compactness
  • 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 is compact if and only if every infinite subset of the space has a complete accumulation point.
Metric spaces
Hausdorff spaces
  • A compact subset of a Hausdorff space is closed.[7] More generally, compact sets can be separated by open sets: if K1 and K2 are compact and disjoint, there exist disjoint open sets U1 and U2 such that K_1 \subset U_1 and K_2 \subset U_2.
  • 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. (Gelfand–Naimark theorem) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.
  • A compact Hausdorff space is normal.
  • Every continuous map from a compact space to a Hausdorff space is closed and proper (i.e., the pre-image of a compact set is compact.) In particular, every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.[8]
  • A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

[edit] 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.

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 given by 2[0,1], with the product topology (Scarborough & Stone 1966, Example 5.3).

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 which (by most definitions) is strictly weaker than compactness is local compactness.

Generalizations of compactness include H-closed and the property of being an H-set in a parent space. A space is H-closed if every open cover has a finite subfamily whose union is dense. Whereas we say X is an H-set of Z if every cover of X with open sets of Z has a finite subfamily whose Z closure contains X.

[edit] See also

[edit] Notes

  1. ^ A space is compact if and only if the space has the finite intersection property on PlanetMath
  2. ^ Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.2; See also Compactness is preserved under a continuous map on PlanetMath
  3. ^ Arkhangel'skii & Fedorchuk 1990, Corollary 5.2.1
  4. ^ Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.3; Closed set in a compact space is compact on PlanetMath; Closed subsets of a compact set are compact on PlanetMath
  5. ^ (Steen & Seebach 1995, p. 67)
  6. ^ Arkhangel'skii & Fedorchuk 1990, Theorem 5.3.7
  7. ^ Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.4
  8. ^ Arkhangel'skii & Fedorchuk 1990, Corollary 5.2.2

[edit] References

[edit] External links

This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the GFDL.

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