Compact space

"Compactness" redirects here. For other uses, seeCompactness (disambiguation).

Inmathematics,specificallygeneral topology,compactnessis a property that seeks to generalize the notion of aclosedandboundedsubset ofEuclidean space.The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes alllimiting valuesof 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$\mathbb {Q}$is not compact, because it has infinitely many "punctures" corresponding to theirrational numbers,and the space ofreal numbers$\mathbb {R}$is not compact either, because it excludes the two limiting values$+\infty$and$-\infty$.However, theextendedreal number linewouldbe 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 beequivalentin othertopological spaces.

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.

One such generalization is that a topological space issequentiallycompactif everyinfinite sequenceof points sampled from the space has an infinitesubsequencethat converges to some point of the space.TheBolzano–Weierstrass theoremstates 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$\mathbb {R} ^{1}$(the real number line), the sequence of points0,  1,  2,  3, ...has no subsequence that converges to any real number.

Compactness was formally introduced byMaurice Fréchetin 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points tospaces of functions.TheArzelà–Ascoli theoremand thePeano existence theoremexemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, includingsequential compactnessandlimit point compactness,were developed in generalmetric spaces.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 setsthat "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 AlexandrovandPavel Urysohnin 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 setis sometimes used as a synonym for compact space, but also often refers to acompact subspaceof atopological space.

Historical development

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.

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated forspaces of functionsrather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations ofGiulio AscoliandCesare Arzelà. 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 convergentsequence 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 HilbertandErhard Schmidt. For a certain class ofGreen's functionscoming 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 operatoras an offshoot of the general notion of a compact space. It wasMaurice Fréchetwho, in1906,had distilled the essence of the Bolzano–Weierstrass property and coined the termcompactnessto refer to this general phenomenon (he used the term already in his 1904 paperwhich 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 Heineshowed that acontinuous functiondefined 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 informationabout 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 AlexandrovandPavel 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.

Basic examples

Anyfinite spaceis 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 pointamong 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 theboundarypoints of the interval, since thelimit pointsmust 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, closeddisksare 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 pointwithinthe 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 ofEuclidean spacein particular is called compact if it isclosedandbounded.This implies, by theBolzano–Weierstrass theorem,that any infinitesequencefrom the set has asubsequencethat converges to a point in the set. Various equivalent notions of compactness, such assequential compactnessandlimit point compactness,can be developed in generalmetric spaces.

In contrast, the different notions of compactness are not equivalent in generaltopological spaces,and the most useful notion of compactness – originally calledbicompactness– is defined usingcoversconsisting ofopen sets(seeOpen cover definitionbelow). 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 aneighbourhoodof 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.

Open cover definition

Formally, atopological spaceXis calledcompactif everyopen coverofXhas afinitesubcover.That is,Xis compact if for every collectionCof open subsetsofXsuch that

$$X=\bigcup _{S\in C}S\,$$

there is afinitesubcollectionF⊆Csuch that

$$X=\bigcup _{S\in F}S\.$$

Some branches of mathematics such asalgebraic geometry,typically influenced by the French school ofBourbaki,use the termquasi-compactfor the general notion, and reserve the termcompactfor topological spaces that are bothHausdorffandquasi-compact.A compact set is sometimes referred to as acompactum,pluralcompacta.

Compactness of subsets

A subsetKof a topological spaceXis said to be compact if it is compact as a subspace (in thesubspace topology). That is,Kis compact if for every arbitrary collectionCof open subsets ofXsuch that

$$K\subseteq \bigcup _{S\in C}S\,$$

there is afinitesubcollectionF⊆Csuch that

$$K\subseteq \bigcup _{S\in F}S\.$$

Because compactness is atopological property,the compactness of a subset depends only on the subspace topology induced on it. It follows that, if$K\subset Z\subset Y$,with subsetZequipped with the subspace topology, thenKis compact inZif and only ifKis compact inY.

Characterization

IfXis a topological space then the following are equivalent:

1. Xis compact; i.e., everyopen coverofXhas a finitesubcover.
2. Xhas 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).
3. XisLindelöfandcountably compact.
4. Any collection of closed subsets ofXwith thefinite intersection propertyhas nonempty intersection.
5. EverynetonXhas a convergent subnet (see the article onnetsfor a proof).
6. EveryfilteronXhas a convergent refinement.
7. Every net onXhas a cluster point.
8. Every filter onXhas a cluster point.
9. EveryultrafilteronXconverges to at least one point.
10. Every infinite subset ofXhas acomplete accumulation point.
11. For every topological spaceY,the projection$X\times Y\to Y$is aclosed mapping(seeproper map).
12. Every open cover linearly ordered by subset inclusion containsX.

Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).

Euclidean space

For anysubsetAofEuclidean space,Ais compact if and only if it isclosedandbounded;this is theHeine–Borel theorem.

As aEuclidean spaceis 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 closedintervalor closedn-ball.

Metric spaces

For any metric space(X,d),the following are equivalent (assumingcountable choice):

1. (X,d)is compact.
2. (X,d)iscompleteandtotally bounded(this is also equivalent to compactness foruniform spaces).
3. (X,d)is sequentially compact; that is, everysequenceinXhas a convergent subsequence whose limit is inX(this is also equivalent to compactness forfirst-countableuniform spaces).
4. (X,d)islimit point compact(also called weakly countably compact); that is, every infinite subset ofXhas at least onelimit pointinX.
5. (X,d)iscountably compact;that is, every countable open cover ofXhas a finite subcover.
6. (X,d)is an image of a continuous function from theCantor set.
7. Every decreasing nested sequence of nonempty closed subsetsS₁⊇S₂⊇...in(X,d)has a nonempty intersection.
8. 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:

1. Lebesgue's number lemma:For every open cover ofX,there exists a numberδ> 0such that every subset ofXof diameter <δis contained in some member of the cover.
2. (X,d)issecond-countable,separableandLindelöf– 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.
3. Xis 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 lineequipped with thediscrete metricis closed and bounded but not compact, as the collection of allsingletonsof the space is an open cover which admits no finite subcover. It is complete but not totally bounded.

Ordered spaces

For an ordered space(X,<)(i.e. a totally ordered set equipped with the order topology), the following are equivalent:

1. (X,<)is compact.
2. Every subset ofXhas a supremum (i.e. a least upper bound) inX.
3. Every subset ofXhas an infimum (i.e. a greatest lower bound) inX.
4. Every nonempty closed subset ofXhas 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.):

1. Every sequence in(X,<)has a subsequence that converges in(X,<).
2. Every monotone increasing sequence inXconverges to a unique limit inX.
3. Every monotone decreasing sequence inXconverges to a unique limit inX.
4. Every decreasing nested sequence of nonempty closed subsetsS₁⊇S₂⊇... in(X,<)has a nonempty intersection.
5. Every increasing nested sequence of proper open subsetsS₁⊆S₂⊆... in(X,<)fails to coverX.

Characterization by continuous functions

LetXbe a topological space andC(X)the ring of real continuous functions onX. For eachp∈X,the evaluation map$\operatorname {ev} _{p}\colon C(X)\to \mathbb {R}$ given byev_p(f) =f(p)is a ring homomorphism. Thekernelofev_pis amaximal ideal,since theresidue fieldC(X)/ker ev_pis the field of real numbers, by thefirst isomorphism theorem.A topological spaceXispseudocompactif 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.There are pseudocompact spaces that are not compact, though.

In general, for non-pseudocompact spaces there are always maximal idealsminC(X)such that the residue fieldC(X)/mis a (non-Archimedean)hyperreal field.The framework ofnon-standard analysisallows for the following alternative characterization of compactness:a topological spaceXis compact if and only if every pointxof the natural extension*Xisinfinitely closeto a pointx₀ofX(more precisely,xis contained in themonadofx₀).

Hyperreal definition

A spaceXis compact if itshyperreal extension*X(constructed, for example, by theultrapower construction) has the property that every point of*Xis 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.

Sufficient conditions

• A closed subset of a compact space is compact.
• A finiteunionof compact sets is compact.
• Acontinuousimage of a compact space is compact.
• The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed);
  • IfXis not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).
• Theproductof any collection of compact spaces is compact. (This isTychonoff's theorem,which is equivalent to theaxiom of choice.)
• In ametrizable space,a subset is compact if and only if it issequentially compact(assumingcountable choice)
• A finite set endowed with any topology is compact.

Properties of compact spaces

• A compact subset of aHausdorff spaceXis closed.
  • IfXis not Hausdorff then a compact subset ofXmay fail to be a closed subset ofX(see footnote for example).
  • IfXis not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).
• In anytopological vector space(TVS), a compact subset iscomplete.However, every non-Hausdorff TVS contains compact (and thus complete) subsets that arenotclosed.
• IfAandBare disjoint compact subsets of a Hausdorff spaceX,then there exist disjoint open setsUandVinXsuch thatA⊆UandB⊆V.
• A continuous bijection from a compact space into a Hausdorff space is ahomeomorphism.
• A compact Hausdorff space isnormalandregular.
• If a spaceXis compact and Hausdorff, then no finer topology onXis compact and no coarser topology onXis Hausdorff.
• If a subset of a metric space(X,d)is compact then it isd-bounded.

Functions and compact spaces

Since acontinuousimage of a compact space is compact, theextreme value theoremholds for such spaces: 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 aproper mapis compact.

Compactifications

Every topological spaceXis an opendense subspaceof a compact space having at most one point more thanX,by theAlexandroff one-point compactification. By the same construction, everylocally compactHausdorff spaceXis an open dense subspace of a compact Hausdorff space having at most one point more thanX.

Ordered compact spaces

A nonempty compact subset of thereal numbershas a greatest element and a least element.

LetXbe asimply orderedset endowed with theorder topology. ThenXis compact if and only ifXis acomplete lattice(i.e. all subsets have suprema and infima).

Examples

• Anyfinite topological space,including theempty set,is compact. More generally, any space with afinite topology(only finitely many open sets) is compact; this includes in particular thetrivial topology.
• Any space carrying thecofinite topologyis compact.
• Anylocally compactHausdorff 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$\mathbb {R}$is homeomorphic to the circleS¹;the one-point compactification of$\mathbb {R} ^{2}$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.
• Theright order topologyorleft order topologyon any boundedtotally ordered setis compact. In particular,Sierpiński spaceis compact.
• Nodiscrete spacewith an infinite number of points is compact. The collection of allsingletonsof the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
• In$\mathbb {R}$carrying thelower limit topology,no uncountable set is compact.
• In thecocountable topologyon an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is notlocally compactbut is stillLindelöf.
• The closedunit interval[0, 1]is compact. This follows from theHeine–Borel theorem.The open interval(0, 1)is not compact: theopen cover${\textstyle \left({\frac {1}{n}},1-{\frac {1}{n}}\right)}$forn= 3, 4, ... does not have a finite subcover. Similarly, the set ofrational numbersin the closed interval[0,1]is not compact: the sets of rational numbers in the intervals${\textstyle \left[0,{\frac {1}{\pi }}-{\frac {1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]}$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$\mathbb {R}$.
• The set$\mathbb {R}$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) ,wherentakes all integer values inZ,cover$\mathbb {R}$but there is no finite subcover.
• On the other hand, theextended real number linecarrying the analogous topologyiscompact; note that the cover described above would never reach the points at infinity and thus wouldnotcover the extended real line. In fact, the set has thehomeomorphismto [−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-sphereis compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensionalnormed vector spaceis compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if itsclosed unit ballis 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 setis compact. In fact, every compact metric space is a continuous image of the Cantor set.
• Consider the setKof all functionsf:$\mathbb {R}$→ [0, 1]from the real number line to the closed unit interval, and define a topology onKso that a sequence$\{f_{n}\}$inKconverges towardsf∈Kif and only if$\{f_{n}(x)\}$converges towardsf(x)for all real numbersx.There is only one such topology; it is called the topology ofpointwise convergenceor theproduct topology.ThenKis 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 setKof all functionsf:[0, 1]→[0, 1]satisfying theLipschitz condition|f(x) −f(y)| ≤ |x−y|for allx,y∈[0,1].Consider onKthe metric induced by theuniform distance$d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.$Then by the Arzelà–Ascoli theorem the spaceKis compact.
• Thespectrumof anybounded linear operatoron aBanach spaceis a nonempty compact subset of thecomplex numbers$\mathbb {C}$.Conversely, any compact subset of$\mathbb {C}$arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space$\ell ^{2}$may have any compact nonempty subset of$\mathbb {C}$as spectrum.
• The space of Borelprobability measureson a compact Hausdorff space is compact for thevague topology,by the Alaoglu theorem.
• A collection of probability measures on the Borel sets of Euclidean space is calledtightif, 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.

Algebraic examples

• Topological groupssuch as anorthogonal groupare compact, while groups such as ageneral linear groupare not.
• Since thep-adic integersarehomeomorphicto the Cantor set, they form a compact set.
• Anyglobal fieldKis a discrete additive subgroup of itsadele ring,and the quotient space is compact. This was used inJohn Tate'sthesisto allowharmonic analysisto be used innumber theory.
• Thespectrumof anycommutative ringwith 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.
• The spectrum of aBoolean algebrais compact, a fact which is part of theStone representation theorem.Stone spaces,compacttotally disconnectedHausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study ofprofinite groups.
• Thestructure spaceof a commutative unitalBanach algebrais a compact Hausdorff space.
• TheHilbert cubeis compact, again a consequence of Tychonoff's theorem.
• Aprofinite group(e.g.Galois group) is compact.

See also

• Compactly generated space
• Compactness theorem
• Eberlein compactum
• Exhaustion by compact sets
• Lindelöf space
• Metacompact space
• Noetherian topological space
• Orthocompact space
• Paracompact space
• Quasi-compact morphism
• Precompact set- also calledtotally bounded
• Relatively compact subspace
• Totally bounded

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External links

• Sundström, Manya Raman (2010). "A pedagogical history of compactness".arXiv:1006.4131v1[math.HO].

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