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In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (i.e., containing all its limit points) and bounded (i.e., having all its points lie within some fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways. One such generalization is that a topological space is ''sequentially'' compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the ''closed'' unit interval , some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence accumulate to 0 (while others accumulate to 1). The same set of points would not accumulate to any point of the ''open'' unit interval ; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points , which is not bounded, has no subsequence that converges to any real number. The concept of a compact space was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem to spaces of functions, rather than geometrical points. Applications of compactness to classical analysis, such as the Arzelà–Ascoli theorem and the Peano existence theorem are of this kind. Following the initial introduction of the concept, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces. In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term ''compactness'', is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character. The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well.

Historical development

Basic examples

Any finite space is trivially compact. A non-trivial example of a compact space is the (closed) unit interval of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point in that interval. For instance, the odd-numbered terms of the sequence get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval of the real numbers is not compact. It is also crucial that the interval be bounded, since in the interval , one could choose the sequence of points , of which no sub-sequence ultimately gets arbitrarily close to any given real number. In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary—without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point ''within'' the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

Definitions

Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces. In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness—originally called ''bicompactness''—is defined using covers consisting of open sets (see ''Open cover definition'' below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally—in a neighbourhood of each point of the space—and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.

Open cover definition

Formally, a topological space is called ''compact'' if each of its open covers has a finite subcover. That is, is compact if for every collection of open subsets of such that :$X = \bigcup_x$, there is a finite subset of such that :$X = \bigcup_x.$ Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both Hausdorff and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.

Compactness of subsets

A subset of a topological space is said to be compact if it is compact as a subspace (in the subspace topology). That is, is compact if for every arbitrary collection of open subsets of such that :$K \subseteq \bigcup_ c$, there is a finite subset of such that :$K \subseteq \bigcup_ c$. Compactness is a "topological" property. That is, if $K \subset Z \subset Y$, with subset equipped with the subspace topology, then is compact in if and only if is compact in .

Equivalent definitions

If is a topological space then the following are equivalent: # is compact. # Every open cover of has a finite subcover. # has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (Alexander's sub-base theorem) # is Lindelöf and countably compact # Any collection of closed subsets of with the finite intersection property has nonempty intersection. # Every net on has a convergent subnet (see the article on nets for a proof). # Every filter on has a convergent refinement. # Every net on has a cluster point. # Every filter on has a cluster point. # Every ultrafilter on converges to at least one point. # Every infinite subset of has a complete accumulation point.

Euclidean space

For any subset of Euclidean space''n'', is compact if and only if it is closed and bounded; this is the Heine–Borel theorem. As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed -ball.

Metric spaces

For any metric space , the following are equivalent (assuming countable choice): # is compact. # is complete and totally bounded (this is also equivalent to compactness for uniform spaces). # is sequentially compact; that is, every sequence in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable uniform spaces). # is limit point compact (also called countably compact); that is, every infinite subset of has at least one limit point in . # is an image of a continuous function from the Cantor set. A compact metric space also satisfies the following properties: # Lebesgue's number lemma: For every open cover of , there exists a number such that every subset of of diameter < ''δ'' is contained in some member of the cover. # is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact. # is closed and bounded (as a subset of any metric space whose restricted metric is ). The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.

Characterization by continuous functions

Let be a topological space and the ring of real continuous functions on . For each , the evaluation map $\operatorname_p\colon C\left(X\right)\to \mathbf$ given by is a ring homomorphism. The kernel of is a maximal ideal, since the residue field is the field of real numbers, by the first isomorphism theorem. A topological space is pseudocompact if and only if every maximal ideal in has residue field the real numbers. For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism. There are pseudocompact spaces that are not compact, though. In general, for non-pseudocompact spaces there are always maximal ideals in such that the residue field is a (non-Archimedean) hyperreal field. The framework of non-standard analysis allows for the following alternative characterization of compactness: a topological space is compact if and only if every point of the natural extension is infinitely close to a point of (more precisely, is contained in the monad of ).

Hyperreal definition

A space is compact if its hyperreal extension (constructed, for example, by the ultrapower construction) has the property that every point of is infinitely close to some point of . For example, an open real interval is not compact because its hyperreal extension contains infinitesimals, which are infinitely close to 0, which is not a point of .

Sufficient conditions

* A closed subset of a compact space is compact. * A finite union of compact sets is compact. * A continuous image of a compact space is compact. * The intersection of any collection of compact subsets of a Hausdorff space is compact (and closed); ** If is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).Let , , and . Endow with the topology generated by the following basic open sets: every subset of is open; the only open sets containing are and ; and the only open sets containing are and . Then and are both compact subsets but their intersection, which is , is not compact. Note that both and are compact open subsets, neither one of which is closed. * The product of any collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.) * In a metrizable space, a subset is compact if and only if it is sequentially compact (assuming countable choice) * A finite set endowed with any topology is compact.

Properties of compact spaces

* A compact subset of a Hausdorff space is closed. ** If is not Hausdorff then a compact subset of may fail to be a closed subset of (see footnote for example).Let and endow with the topology . Then is a compact set but it is not closed. ** If is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).Let be the set of non-negative integers. We endow with the particular point topology by defining a subset to be open if and only if . Then is compact, the closure of is all of , but is not compact since the collection of open subsets does not have a finite subcover. * In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are ''not'' closed. * If and are disjoint compact subsets of a Hausdorff space , then there exist disjoint open set and in such that and . * A continuous bijection from a compact space into a Hausdorff space is a homeomorphism. * A compact Hausdorff space is normal and regular. * If a space is compact and Hausdorff, then no finer topology on is compact and no coarser topology on is Hausdorff. * If a subset of a metric space is compact then it is -bounded.

Functions and compact spaces

Since a continuous image of a compact space is compact, the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum. (Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a proper map is compact.

Compactifications

Every topological space is an open dense subspace of a compact space having at most one point more than , by the Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space is an open dense subspace of a compact Hausdorff space having at most one point more than .

Ordered compact spaces

A nonempty compact subset of the real numbers has a greatest element and a least element. Let be a simply ordered set endowed with the order topology. Then is compact if and only if is a complete lattice (i.e. all subsets have suprema and infima).

Examples

* Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology. * Any space carrying the cofinite topology is compact. * 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 is homeomorphic to the circle ; the one-point compactification of is homeomorphic to the sphere . Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space. * The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpiński space is compact. * No discrete space with an infinite number of points is compact. The collection of all singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact. * In carrying the lower limit topology, no uncountable set is compact. * In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not locally compact but is still Lindelöf. * The closed unit interval is compact. This follows from the Heine–Borel theorem. The open interval is not compact: the open cover $\left \left( \frac, 1 - \frac \right \right)$ for does not have a finite subcover. Similarly, the set of ''rational numbers'' in the closed interval is not compact: the sets of rational numbers in the intervals

Algebraic examples

* Compact groups such as an orthogonal group are compact, while groups such as a general linear group are not. * Since the -adic integers are homeomorphic to the Cantor set, they form a compact set. * The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact, but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact schemes, "quasi" referring to the non-Hausdorff nature of the topology. * The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of profinite groups. * The structure space of a commutative unital Banach algebra is a compact Hausdorff space. * The Hilbert cube is compact, again a consequence of Tychonoff's theorem. * A profinite group (e.g. Galois group) is compact.

* Compactly generated space * Compactness theorem * Eberlein compactum * Exhaustion by compact sets * Lindelöf space * Metacompact space * Noetherian topological space * Orthocompact space * Paracompact space * Precompact set - also called ''totally bounded'' * Relatively compact subspace * Totally bounded

Notes

References

Bibliography

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