Per the compactness criteria for Euclidean space as stated in theHeine–Borel theorem, the intervalA = (−∞, −2] is not compact because it is not bounded. The intervalC = (2, 4) is not compact because it is not closed (but bounded). The intervalB = [0, 1] is compact because it is both closed and bounded.
Inmathematics, specificallygeneral topology,compactness is a property that seeks to generalize the notion of aclosed andbounded subset ofEuclidean space.[1] The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes alllimiting values of points. For example, the openinterval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space ofrational numbers is not compact, because it has infinitely many "punctures" corresponding to theirrational numbers, and the space ofreal numbers is not compact either, because it excludes the two limiting values and. However, theextended real number linewould be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in ametric space, but may not beequivalent in othertopological spaces.
One such generalization is that a topological space issequentially compact if everyinfinite sequence of points sampled from the space has an infinitesubsequence that converges to some point of the space.[2] TheBolzano–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 closedunit interval[0, 1], some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence1/2,4/5,1/3,5/6,1/4,6/7, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval(0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering (the real number line), the sequence of points 0, 1, 2, 3, ... has no subsequence that converges to any real number.
Compactness was formally introduced byMaurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points tospaces of functions. TheArzelà–Ascoli theorem and thePeano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, includingsequential compactness andlimit point compactness, were developed in generalmetric spaces.[3] In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified termcompactness — is phrased in terms of the existence of finite families ofopen 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 byPavel Alexandrov andPavel Urysohn in 1929, exhibits compact spaces as generalizations offinite sets. In spaces that are compact in this sense, it is often possible to patch together information that holdslocally – that is, in a neighborhood of each point – into corresponding statements that hold throughout the space, and many theorems are of this character.
The termcompact set is sometimes used as a synonym for compact space, but also often refers to acompact subspace of atopological space.
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand,Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called alimit point. Bolzano's proof relied on themethod of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts – until it closes down on the desired limit point. The full significance ofBolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered byKarl Weierstrass.[4]
In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated forspaces of functions rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations ofGiulio Ascoli andCesare Arzelà.[5] The culmination of their investigations, theArzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to families ofcontinuous functions, the precise conclusion of which was that it was possible to extract auniformly convergent sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area ofintegral equations, as investigated byDavid Hilbert andErhard Schmidt. For a certain class ofGreen's functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense ofmean convergence – or convergence in what would later be dubbed aHilbert space. This ultimately led to the notion of acompact operator as an offshoot of the general notion of a compact space. It wasMaurice Fréchet who, in1906, had distilled the essence of the Bolzano–Weierstrass property and coined the termcompactness to refer to this general phenomenon (he used the term already in his 1904 paper[6] which led to the famous 1906 thesis).
However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of thecontinuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870,Eduard Heine showed that acontinuous function defined on a closed and bounded interval was in factuniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized byÉmile Borel (1895), and it was generalized to arbitrary collections of intervals byPierre Cousin (1895) andHenri Lebesgue (1904). TheHeine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
This property was significant because it allowed for the passage fromlocal information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed byLebesgue (1904), who also exploited it in the development of theintegral now bearing his name. Ultimately, the Russian school ofpoint-set topology, under the direction ofPavel Alexandrov andPavel Urysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of atopological space.Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called (relative)sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.
Anyfinite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed)unit interval[0,1] ofreal numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be someaccumulation point among these points in that interval. For instance, the odd-numbered terms of the sequence1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, ... get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including theboundary points of the interval, since thelimit 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 bebounded, since in the interval[0,∞), one could choose the sequence of points0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number.
In two dimensions, closeddisks 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 pointwithin 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.
In contrast, the different notions of compactness are not equivalent in generaltopological spaces, and the most useful notion of compactness – originally calledbicompactness – is defined usingcovers consisting ofopen sets (seeOpen cover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as theHeine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is knownlocally – in aneighbourhood 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 isuniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.
Some branches of mathematics such asalgebraic geometry, typically influenced by the French school ofBourbaki, use the termquasi-compact for the general notion, and reserve the termcompact for topological spaces that are bothHausdorff andquasi-compact. A compact set is sometimes referred to as acompactum, pluralcompacta.
A subsetK of a topological spaceX is said to be compact if it is compact as a subspace (in thesubspace topology). That is,K is compact if for every arbitrary collectionC of open subsets ofX such that
there is afinite subcollectionF ⊆C such that
Because compactness is atopological property, the compactness of a subset depends only on the subspace topology induced on it. It follows that, if, with subsetZ equipped with the subspace topology, thenK is compact inZ if and only ifK is compact inY.
As aEuclidean 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 closedinterval or closedn-ball.
(X,d) is sequentially compact; that is, everysequence inX has a convergent subsequence whose limit is inX (this is also equivalent to compactness forfirst-countableuniform spaces).
(X,d) islimit point compact (also called weakly countably compact); that is, every infinite subset ofX has at least onelimit point inX.
(X,d) iscountably compact; that is, every countable open cover ofX has a finite subcover.
(X,d) is an image of a continuous function from theCantor set.[15]
Every decreasing nested sequence of nonempty closed subsetsS1 ⊇S2 ⊇ ... in(X,d) has a nonempty intersection.
Every increasing nested sequence of proper open subsetsS1 ⊆S2 ⊆ ... in(X,d) fails to coverX.
A compact metric space(X,d) also satisfies the following properties:
Lebesgue's number lemma: For every open cover ofX, there exists a numberδ > 0 such that every subset ofX of diameter <δ is contained in some member of the cover.
(X,d) issecond-countable,separable andLindelöf – 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.
X is closed and bounded (as a subset of any metric space whose restricted metric isd). The converse may fail for a non-Euclidean space; e.g. thereal line equipped with thediscrete metric is closed and bounded but not compact, as the collection of allsingletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.
For an ordered space(X, <) (i.e. a totally ordered set equipped with the order topology), the following are equivalent:
(X, <) is compact.
Every subset ofX has a supremum (i.e. a least upper bound) inX.
Every subset ofX has an infimum (i.e. a greatest lower bound) inX.
Every nonempty closed subset ofX has a maximum and a minimum element.
An ordered space satisfying (any one of) these conditions is called a complete lattice.
In addition, the following are equivalent for all ordered spaces(X, <), and (assumingcountable choice) are true whenever(X, <) is compact. (The converse in general fails if(X, <) is not also metrizable.):
Every sequence in(X, <) has a subsequence that converges in(X, <).
Every monotone increasing sequence inX converges to a unique limit inX.
Every monotone decreasing sequence inX converges to a unique limit inX.
Every decreasing nested sequence of nonempty closed subsetsS1 ⊇S2 ⊇ ... in(X, <) has a nonempty intersection.
Every increasing nested sequence of proper open subsetsS1 ⊆S2 ⊆ ... in(X, <) fails to coverX.
LetX be a topological space andC(X) the ring of real continuous functions onX. For eachp ∈X, the evaluation mapgiven byevp(f) =f(p) is a ring homomorphism. Thekernel ofevp is amaximal ideal, since theresidue fieldC(X)/ker evp is the field of real numbers, by thefirst isomorphism theorem. A topological spaceX ispseudocompact if and only if every maximal ideal inC(X) has residue field the real numbers. Forcompletely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.[16] There are pseudocompact spaces that are not compact, though.
In general, for non-pseudocompact spaces there are always maximal idealsm inC(X) such that the residue fieldC(X)/m is a (non-Archimedean)hyperreal field. The framework ofnon-standard analysis allows for the following alternative characterization of compactness:[17] a topological spaceX is compact if and only if every pointx of the natural extension*X isinfinitely close to a pointx0 ofX (more precisely,x is contained in themonad ofx0).
A spaceX is compact if itshyperreal extension*X (constructed, for example, by theultrapower construction) has the property that every point of*X is infinitely close to some point ofX ⊂*X. For example, an open real intervalX = (0, 1) is not compact because its hyperreal extension*(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point ofX.
IfX is not Hausdorff then a compact subset ofX may fail to be a closed subset ofX (see footnote for example).[b]
IfX is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).[c]
In anytopological vector space (TVS), a compact subset iscomplete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that arenot closed.
IfA andB are disjoint compact subsets of a Hausdorff spaceX, then there exist disjoint open setsU andV inX such thatA ⊆U andB ⊆V.
A continuous bijection from a compact space into a Hausdorff space is ahomeomorphism.
Since acontinuous image of a compact space is compact, theextreme value theorem holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.[20] (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 aproper map is compact.
Every topological spaceX is an opendense subspace of a compact space having at most one point more thanX, by theAlexandroff one-point compactification. By the same construction, everylocally compact Hausdorff spaceX is an open dense subspace of a compact Hausdorff space having at most one point more thanX.
Anylocally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means ofAlexandroff one-point compactification. The one-point compactification of is homeomorphic to the circleS1; the one-point compactification of is homeomorphic to the sphereS2. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
Nodiscrete space with an infinite number of points is compact. The collection of allsingletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
The closedunit interval[0, 1] is compact. This follows from theHeine–Borel theorem. The open interval(0, 1) is not compact: theopen cover forn = 3, 4, ... does not have a finite subcover. Similarly, the set ofrational numbers in the closed interval[0,1] is not compact: the sets of rational numbers in the intervals cover all the rationals in [0, 1] forn = 4, 5, ... but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of .
The set 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), wheren takes all integer values inZ, cover but there is no finite subcover.
On the other hand, theextended real number line carrying the analogous topologyis compact; note that the cover described above would never reach the points at infinity and thus wouldnot cover the extended real line. In fact, the set has thehomeomorphism to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
For everynatural numbern, then-sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensionalnormed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if itsclosed unit ball is compact.
On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. (Alaoglu's theorem)
TheCantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set.
Consider the setK of all functionsf : → [0, 1] from the real number line to the closed unit interval, and define a topology onK so that a sequence inK converges towardsf ∈K if and only if converges towardsf(x) for all real numbersx. There is only one such topology; it is called the topology ofpointwise convergence or theproduct topology. ThenK is a compact topological space; this follows from theTychonoff theorem.
A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded (Arzelà–Ascoli theorem).
Consider the setK of all functionsf :[0, 1] →[0, 1] satisfying theLipschitz condition|f(x) − f(y)| ≤ |x − y| for allx, y ∈ [0,1]. Consider onK the metric induced by theuniform distance Then by the Arzelà–Ascoli theorem the spaceK is compact.
Thespectrum of anybounded linear operator on aBanach space is a nonempty compact subset of thecomplex numbers. Conversely, any compact subset of arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of as spectrum.
A collection of probability measures on the Borel sets of Euclidean space is calledtight if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures. Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.
Thespectrum of anycommutative ring with theZariski topology (that is, the set of all prime ideals) is compact, but neverHausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compactschemes, "quasi" referring to the non-Hausdorff nature of the topology.
^LetX = {a,b} ∪,U = {a} ∪, andV = {b} ∪. EndowX with the topology generated by the following basic open sets: every subset of is open; the only open sets containinga areX andU; and the only open sets containingb areX andV. ThenU andV are both compact subsets but their intersection, which is, is not compact. Note that bothU andV are compact open subsets, neither one of which is closed.
^LetX = {a,b} and endowX with the topology{X, ∅, {a}}. Then{a} is a compact set but it is not closed.
^LetX be the set of non-negative integers. We endowX with theparticular point topology by defining a subsetU ⊆X to be open if and only if0 ∈U. ThenS := {0} is compact, the closure ofS is all ofX, butX is not compact since the collection of open subsets{{0,x} :x ∈X} does not have a finite subcover.
^Frechet, M. 1904."Generalisation d'un theorem de Weierstrass".Analyse Mathematique.
^Weisstein, Eric W."Compact Space".Wolfram MathWorld. Retrieved2019-11-25.
^Here, "collection" means "set" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset".
Alexandrov, Pavel;Urysohn, Pavel (1929). "Mémoire sur les espaces topologiques compacts".Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Mathematical Sciences.14.
Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.).General Topology I. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer.ISBN978-0-387-18178-3..
Arzelà, Cesare (1895). "Sulle funzioni di linee".Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat.5 (5):55–74.
Arzelà, Cesare (1882–1883). "Un'osservazione intorno alle serie di funzioni".Rend. Dell' Accad. R. Delle Sci. dell'Istituto di Bologna:142–159.
Ascoli, G. (1883–1884). "Le curve limiti di una varietà data di curve".Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat.18 (3):521–586.
