|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{redirect|Compactness|the concept in first-order logic|compactness theorem}}
| |
| In the mathematical discipline of [[general topology]], '''compactness''' is a property that generalizes the notion of a subset of [[Euclidean space]] being [[closed set|closed]] (that is, containing all its [[limit point]]s) and [[bounded set|bounded]] (that is, all points within some fixed distance of each other). This notion is generalized to more general [[topological space]]s in various ways. For instance, a space is [[sequentially compact|''sequentially'' compact]] if any [[infinite sequence]] of points sampled from the space must eventually, infinitely often, get arbitrarily close to some point of the space. The [[Bolzano–Weierstrass theorem]] states that a subset of Euclidean space is compact in this sense if and only if it is closed and bounded. Examples include a [[closed interval]] or a [[rectangle]]. Thus if one chooses an infinite number of points in the ''closed'' [[unit interval]], some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers {{nowrap|1/2, 4/5, 1/3, 5/6, 1/4, 6/7, …}} get arbitrarily close to 0. (Also, some get arbitrarily close to 1.) The same set of points would not have, as a [[limit point]], 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 {{math|0, 1, 2, 3, …}} has no sub-sequence that ultimately gets arbitrarily close to any given real number.
| |
|
| |
|
| Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of [[function space|functions]]. The term ''compact'' was introduced into mathematics by [[Maurice Fréchet]] in 1906 as a distillation of this concept. Compactness in this more general situation plays an extremely important role in [[mathematical analysis]], because many classical and important theorems of 19th century analysis, such as the [[extreme value theorem]], are easily generalized to this situation. A typical application is furnished by the [[Arzelà–Ascoli theorem]] and in particular the [[Peano existence theorem]], in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.
| |
|
| |
|
| Various equivalent notions of compactness, including sequential compactness and [[limit point compact]]ness, can be developed in general [[metric space]]s. In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by [[Pavel Alexandrov]] and [[Pavel Urysohn]] in 1929, involves the existence of certain finite families of [[open set]]s that "[[Cover (topology)|cover]]" the space in the sense that each point of the space must lie in some set contained in the family. The standard unqualified use of the term ''compact'' in mathematics usually means compactness in this latter sense. This more subtle definition exhibits compact spaces as generalizations of [[finite set]]s. In spaces that are compact in this sense, it is often possible to patch together information that holds [[local property|locally]]—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many [[#Theorems|theorems]] are of this character.
| | Beverly Hills Liposuction -- Carrying Out What The Stars Do<br><br>If you're a true meals and consume aficionado, why not live close to the decision dining districts in Los Angeles? With the LA real estate downturn in entire dive manner, numerous value prohibitive places are now considerably more affordable, [http://En.Search.Wordpress.com/?q=Link+Schwartz Link Schwartz] particularly for condos. So LA foodies consider discover: Now's the time to shift even closer to your correct love. Foods, that is.<br><br>One time Gleason posed the query, "What in your in your job is not possible?" In 1964 Frank Sinatra was about the most significant star on the earth and acknowledged for not offering interviews, so King responded that a Sinatra interview would fit the invoice.<br><br>Onwuasor also tied the sport at sport at 60 when he caught a 15-lawn reception in the second OT period. That came after Kevin Porsche gave the Centaurs (four-three, 1-one) a sixty-53 lead on a thirteen-yard capture.<br><br>Brandi is encouraging her supporters to write their viewpoint of her book. Do you program on creating a evaluation for Brandi? What ranking would you give the e-book out of five?<br><br>Mimi storms the glistening glass outlets like Generalissimo Franco. I stick to limply like a Demise March survivor considering if my son weren't in Catholic university I could just reside Link Schwartz in sin.<br><br>Former "Link Schwartz 90210" actress Jennie Garth has been candid about her break up with "Twilight" star Peter Facinelli given that the two parted ways 9 months in the past following eleven many years of marriage. Whilst Facinelli has been considerably less general public about his feelings with regards to the split, the actor recently stepped out with girlfriend and "Loosies" costar Jaimie Alexander, triggering the media lens to be aimed in anticipation at Garth's reaction.<br><br>P. Diddy / Sean Combs / Puffy / Puffy Daddy - whatsoever he's heading by these times, it shouldn't be RAPPER! "I'm the macaroni & the cheese" I feel ample has been said! You would considering hanging out with B.I.G. [http://www.swarthmore.edu/SocSci/bschwar1/ Link K Schwartz] for a long time he would have realized Link Schwartz a tiny some thing. Guess not!<br><br>A number of months soon after Faye moved in to her three bed room, 3.5 bathtub property in the Hollywood Hills, a stalker snuck onto her house and remaining a threatening note. Law enforcement have been notified and on the hunt for the stalker, but thus considerably there is no suspect. Police have included further patrol's to Faye's community.<br><br>But why is Erickson standing there kissing Facinelli's spouse in the climactic scene of Unintentionally in Really like, while her actual-existence partner-who scripted the point-is forced to stand on the sidelines and look at?<br><br>Seeing Priestly on the small monitor is a extremely familiar sight for many individuals, specially these that grew up in the nineties. Though 'A Aspect Purchase of Lifestyle' will not propel Priestley's star to the levels that 90210 did, it is a solid function. Former 90210 supporters must undoubtedly check out the display out, for the nostalgic sensation by itself. |
| | |
| ==Introduction==
| |
| An example of a compact space is the [[unit interval]] {{closed-closed|0,1}} of [[real number]]s. 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 {{math|1, 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 the [[boundary (topology)|boundary]] points of the interval, since the [[limit point]]s 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 set|bounded]], since in the interval {{closed-open|0,∞}} one could choose the sequence of points {{math|0, 1, 2, 3, …}}, of which no sub-sequence ultimately gets arbitrarily close to any given real number.
| |
| | |
| In two dimensions, closed [[Disk (mathematics)|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 tend to the missing point without tending 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.
| |
| | |
| Compactness generalizes many important properties of [[closed (topology)|closed]] and bounded intervals in the real line; that is, intervals of the form {{closed-closed|{{mvar|a}}, {{mvar|b}}}} for real numbers {{mvar|a}} and {{mvar|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.
| |
| | |
| 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 set|closed]] and [[bounded set|bounded]]. This implies, by the [[Bolzano–Weierstrass theorem]], that any infinite [[sequence (mathematics)|sequence]] from the set has a [[subsequence]] that converges to a point in the set. This puts a fine point on the idea of taking "steps" in a space. Various equivalent notions of compactness, such as [[sequential compactness]] and [[limit point compact]]ness, can be developed in general [[metric space]]s.
| |
| | |
| In general [[topological space]]s, however, the different notions of compactness are not equivalent, and the most useful notion of compactness—originally called ''bicompactness''—involves families of [[open set]]s that [[cover (topology)|cover]] the space in the sense that each point of the space must lie in some set contained in the family. Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space. 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 [[local property|locally]]—in a [[neighbourhood (mathematics)|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.
| |
| | |
| ==Definition==
| |
| Formally, a [[topological space]] ''X'' is called ''compact'' if each of its [[open cover]]s has a [[finite set|finite]] [[subcover]]. Otherwise it is called ''non-compact''. Explicitly, this means that for every arbitrary collection
| |
| | |
| :<math>\{U_\alpha\}_{\alpha\in A}</math>
| |
| | |
| of open subsets of {{mvar|X}} such that
| |
| | |
| :<math>X = \bigcup_{\alpha\in A} U_\alpha,</math>
| |
| | |
| there is a finite subset {{mvar|J}} of {{mvar|A}} such that
| |
| | |
| :<math>X = \bigcup_{i\in J} U_i.</math>
| |
| | |
| Some branches of mathematics such as [[algebraic geometry]], typically influenced by the French school of [[Nicolas Bourbaki|Bourbaki]], use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both [[Hausdorff spaces|Hausdorff]] and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''.
| |
| | |
| ===Hyperreal definition===
| |
| A space ''X'' is compact if its natural extension ''*X'' (for example, an ultrapower) has the property that every point of ''*X'' is infinitely close to a suitable point of <math>X\subset{}^\ast X</math>. For example, an open real interval ''X''=(0,1) is not compact because its [[hyperreal number|hyperreal]] extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of ''X''.
| |
| | |
| ===Compactness of subspaces===
| |
| A subset ''K'' of a topological space ''X'' is called compact if it is compact in the [[induced topology]]. Explicitly, this means that for every arbitrary collection
| |
| :<math>\{U_\alpha\}_{\alpha\in A}</math>
| |
| | |
| of open subsets of {{mvar|X}} such that
| |
| | |
| :<math>K\subset\bigcup_{\alpha\in A} U_\alpha,</math>
| |
| | |
| there is a finite subset ''J'' of ''A'' such that
| |
| | |
| :<math>K\subset\bigcup_{i\in J} U_i.</math>
| |
| | |
| == 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]] ([[#CITEREFBolzano1817|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 a [[limit point]]. Bolzano's proof relied on the [[method 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 of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by [[Karl Weierstrass]].<ref>{{harvnb|Kline|1972|pp=952–953}}; {{harvnb|Boyer|Merzbach|1991|p=561}}</ref> | |
| | |
| In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for [[function spaces|spaces 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 of [[Giulio Ascoli]] and [[Cesare Arzelà]].<ref>{{harvnb|Kline|1972|loc=Chapter 46, §2}}</ref> The culmination of their investigations, the [[Arzelà–Ascoli theorem]], was a generalization of the Bolzano–Weierstrass theorem to families of [[continuous function]]s, the precise conclusion of which was that it was possible to extract a [[uniform convergence|uniformly 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 of [[integral equation]]s, as investigated by [[David Hilbert]] and [[Erhard Schmidt]]. For a certain class of [[Green's function|Green functions]] coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of [[mean convergence]]—or convergence in what would later be dubbed a [[Hilbert space]]. This ultimately led to the notion of a [[compact operator]] as an offshoot of the general notion of a compact space. It was [[Maurice René Fréchet|Maurice Fréchet]] who, in [[#CITEREFFréchet1906|1906]], had distilled the essence of the Bolzano–Weierstrass property and coined the term ''compactness'' to refer to this general phenomenon.
| |
| | |
| However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the [[linear continuum|continuum]], which was seen as fundamental for the rigorous formulation of analysis. In 1870, [[Eduard Heine]] showed that a [[continuous function]] defined on a closed and bounded interval was in fact [[uniformly 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 by [[Émile Borel]] ([[#CITEREFBorel1895|1895]]), and it was generalized to arbitrary collections of intervals by [[Pierre Cousin]] (1895) and [[Henri Lebesgue]] ([[#CITEREFLebesgue1904|1904]]). The [[Heine–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 from [[local property|local 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 by {{harvtxt|Lebesgue|1904}}, who also exploited it in the development of the [[Lebesgue integral|integral now bearing his name]]. Ultimately the Russian school of [[point-set topology]], under the direction of [[Pavel Alexandrov]] and [[Pavel Urysohn]], formulated Heine–Borel compactness in a way that could be applied to the modern notion of a [[topological space]]. {{harvtxt|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.
| |
| <!--
| |
| One of the main reasons for studying compact spaces is because they are in some ways very similar to [[finite set]]s: 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 [[separated sets|separate]] ''x'' and ''A'' by [[neighborhood (topology)|neighborhoods]]: for each ''a'' in ''A'', let ''U''(''x'') and ''V''(''a'') be disjoint neighborhoods containing ''x'' and ''a'', respectively. Then the intersection of all the ''U''(''x'') and the union of all the ''V''(''a'') are the required neighborhoods of ''x'' and ''A''.
| |
| | |
| Note that if ''A'' is [[Infinity|infinite]], the proof fails, because the intersection of arbitrarily many neighborhoods of ''x'' might not be a neighborhood of ''x''. The proof can be "rescued", however, if ''A'' is compact: we simply take a finite subcover of the cover {{math|{''V''(''a'') : ''a'' ∈ 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 neighborhoods 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 neighborhoods – 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.
| |
| --> | |
| | |
| == Examples ==
| |
| | |
| === General topology ===
| |
| * Any [[finite topological space]], including the [[empty set]], is compact. Slightly 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 '''R''' is homeomorphic to the circle '''S'''<sup>1</sup>; the one-point compactification of '''R'''<sup>2</sup> is homeomorphic to the sphere '''S'''<sup>2</sup>. 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, [[Sierpinski space]] is compact.
| |
| * '''R''', carrying the [[lower limit topology]], satisfies the property that no uncountable set is compact.
| |
| * In the [[cocountable topology]] on '''R''' (or any uncountable set for that matter), no infinite set is compact.
| |
| * Neither of the spaces in the previous two examples are [[locally compact]] but both are still [[Lindelöf space|Lindelöf]].
| |
| | |
| === Analysis and algebra ===
| |
| * The closed [[unit interval]] {{closed-closed|0,1}} is compact. This follows from the [[Heine–Borel theorem]]. The open interval {{open-open|0,1}} is not compact: the [[open cover]]
| |
| ::<math>\left ( \frac{1}{n}, 1 - \frac{1}{n} \right )</math>
| |
| :for {{math|1={{mvar|n}} = 3, 4, … }} does not have a finite subcover. Similarly, the set of ''[[rational numbers]]'' in the closed interval {{closed-closed|0,1}} is not compact: the sets of rational numbers in the intervals
| |
| ::<math>\left[0,\frac{1}{\pi}-\frac{1}{n}\right] \ \text{and} \ \left[\frac{1}{\pi}+\frac{1}{n},1\right]</math>
| |
| :cover all the rationals in [0, 1] for {{math|1={{mvar|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 {{open-open|{{mvar|n}}−1, {{mvar|n}}+1}} , where {{mvar|n}} takes all integer values in '''Z''', cover '''R''' but there is no finite subcover.
| |
| * More generally, [[compact group]]s such as an [[orthogonal group]] are compact, while groups such as a [[general linear group]] are not.
| |
| * For every [[natural number]] {{mvar|n}}, the {{mvar|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.
| |
| * On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. ([[Alaoglu's theorem]])
| |
| * The [[Cantor set]] is compact. In fact, every compact metric space is a continuous image of the Cantor set.
| |
| * Since the [[p-adic numbers|''p''-adic integers]] are [[homeomorphic]] to the Cantor set, they form a compact set.
| |
| * Consider the set ''K'' of all functions ''f'' : '''R''' → [0,1] from the real number line to the closed unit interval, and define a topology on ''K'' so that a sequence <math>\{f_n\}</math> in ''K'' converges towards <math>f\in K</math> if and only if <math>\{f_n(x)\}</math> converges towards ''f''(''x'') for all real numbers ''x''. There is only one such topology; it is called the [[topology of pointwise convergence]]. Then ''K'' is a compact topological space; this follows from the [[Tychonoff theorem]].
| |
| * Consider the set ''K'' of all functions ''f'' : {{closed-closed|0,1}} → {{closed-closed|0,1}} satisfying the [[Lipschitz condition]] |''f''(''x'') − ''f''(''y'')| ≤ |''x'' − ''y''| for all ''x'', ''y'' ∈ {{closed-closed|0,1}}. Consider on ''K''  the metric induced by the [[uniform convergence|uniform distance]]
| |
| ::<math>d(f,g)=\sup_{x \in [0, 1]} |f(x)-g(x)|.</math>
| |
| :Then by [[Arzelà–Ascoli theorem]] the space ''K'' is compact.
| |
| * The [[spectrum of an operator|spectrum]] of any [[bounded linear operator]] on a [[Banach space]] is a nonempty compact subset of the [[complex number]]s '''C'''. Conversely, any compact subset of '''C''' arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space [[sequence spaces#ℓp spaces|<math>\ell^2</math>]] may have any compact nonempty subset of '''C''' as spectrum.
| |
| * The [[spectrum of a ring|spectrum]] of any [[commutative ring]] with the [[Zariski topology]] (that is, the set of all prime ideals) is compact, but never [[Hausdorff space|Hausdorff]] (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact [[scheme (mathematics)|schemes]], "quasi" referring to the non-Hausdorff nature of the topology.
| |
| * The [[spectrum of a boolean algebra|spectrum]] of a [[Boolean algebra (structure)|Boolean algebra]] is compact, a fact which is part of the [[Stone representation theorem]]. [[Stone space]]s, compact [[totally disconnected space|totally disconnected]] Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of [[profinite group]]s.
| |
| * 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.
| |
| | |
| == Theorems ==
| |
| Some theorems related to compactness (see the [[glossary of topology]] for the definitions):
| |
| | |
| * A [[continuous function (topology)|continuous]] image of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.2}}; See also {{planetmathref|id=4689|title=Compactness is preserved under a continuous map}}</ref>
| |
| * The pre-image of a compact space under a [[proper map]] is compact.
| |
| * The [[extreme value theorem]]: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Corollary 5.2.1}}</ref> (Slightly more generally, this is true for an upper semicontinuous function.)
| |
| * A closed subset of a compact space is compact.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.3}}; {{planetmathref|id=4177|title=Closed set in a compact space is compact}}; {{planetmathref|id=4691|title=Closed subsets of a compact set are compact}}</ref>
| |
| * A finite union of compact sets is compact.
| |
| * A nonempty compact subset of the [[real number]]s has a greatest element and a least element.
| |
| * The [[product topology|product]] of any collection of compact spaces is compact. ([[Tychonoff's theorem]], which is equivalent to the [[axiom of choice]])
| |
| * Every topological space ''X'' is an open [[dense topological subspace|dense subspace]] of a compact space having at most one point more than ''X'', by the [[Compactification (mathematics)|Alexandroff one-point compactification]]. By the same construction, every [[locally compact]] Hausdorff space ''X'' is an open dense subspace of a compact Hausdorff space having at most one point more than ''X''.
| |
| *Let ''X'' be a [[total order|simply ordered]] set endowed with the [[order topology]]. Then ''X'' is compact if and only if ''X'' is a [[complete lattice]] (i.e. all subsets have suprema and infima).<ref>{{harv|Steen|Seebach|1995|p=67}}</ref>
| |
| | |
| ;Characterizations of compactness
| |
| Assuming the [[axiom of choice]], the following are equivalent.
| |
| # A topological space ''X'' is compact.
| |
| # Every [[open cover]] of ''X'' has a finite [[subcover]].
| |
| # ''X'' 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]])
| |
| # Any collection of closed subsets of ''X'' with the [[finite intersection property]] has nonempty intersection.
| |
| # Every [[Net (mathematics)|net]] on ''X'' has a convergent subnet (see the article on [[Net (mathematics)|nets]] for a proof).
| |
| # Every [[mathematical filter|filter]] on ''X'' has a convergent refinement.
| |
| # Every [[ultrafilter]] on ''X'' converges to at least one point.
| |
| # Every infinite subset of ''X'' has a [[complete accumulation point]].<ref>{{harv|Kelley|1955|p=163}}</ref>
| |
| | |
| ;Euclidean space
| |
| For any [[subset]] ''A'' of [[Euclidean space]] '''R'''<sup>''n''</sup>, the following are equivalent:
| |
| # ''A'' is compact.
| |
| # Every [[sequence]] in ''A'' has a [[Limit of a sequence|convergent]] subsequence, whose limit lies in ''A''.
| |
| # Every infinite subset of ''A'' has at least one [[limit point]] in ''A''.
| |
| # ''A'' is [[closed set|closed]] and [[bounded set|bounded]] ([[Heine–Borel theorem]]).
| |
| # ''A'' is [[complete (topology)|complete]] and [[totally bounded]].
| |
| | |
| In practice, the condition (4) is easiest to verify, for example a closed [[interval (mathematics)|interval]] or closed ''n''-ball. Note that, in a metric space, every compact subset is closed and bounded. However, the converse may fail in non-Euclidean '''R'''<sup>''n''</sup>. For example, the [[real line]] equipped with the [[discrete topology]] is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.
| |
| | |
| ;Metric spaces
| |
| * A [[metric space]] (or [[uniform space]]) is compact if and only if it is [[completeness (topology)|complete]] and [[totally bounded]].<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.3.7}}</ref>
| |
| * If the metric space ''X'' is compact and an open cover of ''X'' is given, then there exists a number {{math|δ > 0}} such that every subset of ''X'' of diameter < δ is contained in some member of the cover. ([[Lebesgue's number lemma]])
| |
| * Every compact metric space is [[Separable space|separable]].
| |
| * A metric space (or more generally any [[first-countable]] [[uniform space]]) is compact if and only if every [[sequence]] in the space has a convergent subsequence. ([[Sequentially compact]])
| |
| | |
| ;Hausdorff spaces
| |
| * A compact subset of a [[Hausdorff space]] is closed.<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Theorem 5.2.4}}</ref> More generally, compact sets can be separated by open sets: if ''K''<sub>1</sub> and ''K''<sub>2</sub> are compact and disjoint, there exist disjoint open sets ''U''<sub>1</sub> and ''U''<sub>2</sub> such that <math>K_1 \subset U_1</math> and <math>K_2 \subset U_2</math>. This is to say, compact Hausdorff space is [[normal space|normal]].
| |
| * Two compact Hausdorff spaces ''X''<sub>1</sub> and ''X''<sub>2</sub> are homeomorphic if and only if their [[mathematical ring|rings]] of continuous real-valued functions C(''X''<sub>1</sub>) and C(''X''<sub>2</sub>) are [[ring homomorphism|isomorphic]]. ([[Gelfand–Naimark theorem]]) Properties of the [[Banach space]] of [[continuous functions on a compact Hausdorff space]] are central to abstract analysis.
| |
| * Every continuous map from a compact space to a Hausdorff space is [[closed map|closed]] and [[proper map|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]].<ref>{{harvnb|Arkhangel'skii|Fedorchuk|1990|loc=Corollary 5.2.2}}</ref>
| |
| * A topological space can be embedded in a compact Hausdorff space if and only if it is a [[Tychonoff space]].
| |
| | |
| ;Characterization by continuous functions
| |
| Let ''X'' be a topological space and C(''X'') the ring of real continuous functions on ''X''. For each ''p''∈''X'', the evaluation map
| |
| :<math>\operatorname{ev}_p: C(X)\to \mathbf{R}</math>
| |
| given by ev<sub>''p''</sub>(''f'')=''f''(''p'') is a ring homomorphism. The [[kernel (algebra)|kernel]] of ev<sub>''p''</sub> is a [[maximal ideal]], since the [[residue field]] {{nowrap|C(''X'')/ker ev<sub>''p''</sub>}} is the field of real numbers, by the [[first isomorphism theorem]]. A topological space ''X'' is [[pseudocompact space|pseudocompact]] if and only if every maximal ideal in C(''X'') has residue field the real numbers. For [[completely regular space]]s, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.<ref>{{harvnb|Gillman|Jerison|1976|loc=§5.6}}</ref> There are pseudocompact spaces that are not compact, though.
| |
| | |
| In general, for non-pseudocompact spaces there are always maximal ideals ''m'' in C(''X'') such that the residue field C(''X'')/''m'' is a ([[non-archimedean field|non-archimedean]]) [[hyperreal field]]. The framework of [[non-standard analysis]] allows for the following alternative characterization of compactness:<ref>{{harvnb|Robinson||loc=Theorem 4.1.13}}</ref> a topological space ''X'' is compact if and only if every point ''x'' of the natural extension ''*X'' is [[infinitesimal|infinitely close]] to a point ''x''<sub>0</sub> of ''X'' (more precisely, ''x'' is contained in the [[monad (non-standard analysis)|monad]] of ''x''<sub>0</sub>).
| |
| | |
| == 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 space|Sequentially compact]]: Every [[sequence]] has a convergent subsequence.
| |
| * [[countably compact space|Countably compact]]: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
| |
| * [[Pseudocompact space|Pseudocompact]]: Every real-valued [[continuous function (topology)|continuous]] [[function (mathematics)|function]] on the space is bounded.
| |
| * [[Limit point compact]]: Every infinite subset has a [[limit point]].
| |
| | |
| While all these conditions are equivalent for [[metric space]]s, 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<sup>{{closed-closed|0,1}}</sup>, with the product topology {{harv|Scarborough|Stone|1966|loc=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 space]]s. 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 [[locally compact space|local compactness]].
| |
| | |
| Generalizations of compactness include [[H-closed space|H-closed]] and the property of being an [[H-set]] in a parent space. A Hausdorff 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'''.
| |
| | |
| == See also ==
| |
| * [[Compactly generated space]]
| |
| * [[Eberlein compactum]]
| |
| * [[Exhaustion by compact sets]]
| |
| * [[Lindelöf space]]
| |
| * [[Metacompact space]]
| |
| * [[Noetherian space]]
| |
| * [[Orthocompact space]]
| |
| * [[Paracompact space]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{citation |last1=Alexandrov |first1=Pavel |authorlink1=Pavel Alexandrov |last2=Urysohn |first2=Pavel |authorlink2=Pavel Urysohn |title=Mémoire sur les espaces topologiques compacts |journal=Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the section of mathematical sciences |volume=14 |year=1929}}.
| |
| *{{citation |last1=Arkhangel'skii |first1=A.V. |last2=Fedorchuk |first2=V.V. |contribution=The basic concepts and constructions of general topology |editor1=Arkhangel'skii, A.V. |editor2=Pontrjagin, L.S. |title=General topology I |publisher=Springer |year=1990 |isbn=978-0-387-18178-3 |series=Encyclopedia of the Mathematical Sciences |volume=17}}.
| |
| *{{springer|id=C/c023530|title=Compact space|first=A.V.|last=Arkhangel'skii}}.
| |
| *{{citation |first=Bernard |last=Bolzano |authorlink=Bernard Bolzano |title=Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege |year=1817 |url=http://books.google.com/?id=EoW4AAAAIAAJ&dq=%22Rein%20analytischer%20Beweis%20des%20Lehrsatzes%22&pg=PA2-IA3#v=onepage&q= |publisher=Wilhelm Engelmann}} (''Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation'').
| |
| *{{citation |last=Borel |first=Émile |authorlink=Émile Borel |title=Sur quelques points de la théorie des fonctions |journal=[[Annales Scientifiques de l'École Normale Supérieure]]|series= 3 |volume=12 |year=1895 |pages=9–55 |url=http://www.numdam.org/numdam-bin/item?id=ASENS_1895_3_12__9_0 |jfm=26.0429.03 }}
| |
| *{{Citation | last1=Boyer | first1=Carl B. | author1-link=Carl Benjamin Boyer | title=The history of the calculus and its conceptual development | publisher=[[Dover Publications]] | location=New York | mr=0124178 | year=1959}}.
| |
| * {{citation |first=Cesare |last=Arzelà |authorlink=Cesare Arzelà |title=Sulle funzioni di linee |journal=Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. |volume=5 |issue=5 |pages=55–74 |year=1895}}.
| |
| * {{citation |first=Cesare |last=Arzelà |authorlink=Cesare Arzelà |title=Un'osservazione intorno alle serie di funzioni |journal=Rend. Dell' Accad. R. Delle Sci. Dell'Istituto di Bologna |pages=142–159 |year=1882–1883}}.
| |
| * {{citation |first=G. |last=Ascoli |authorlink=Giulio Ascoli |title=Le curve limiti di una varietà data di curve |journal=Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. |volume=18 |issue=3 |pages=521–586 |year=1883–1884}}.
| |
| *{{citation | first=Maurice |last=Fréchet |authorlink=Maurice Fréchet |title=Sur quelques points du calcul fonctionnel |year=1906 |journal= [[Rendiconti del Circolo Matematico di Palermo]] |volume=22 |doi=10.1007/BF03018603 |pages=1–72 |issue=1}}.
| |
| *{{citation | last1=Gillman|first1=Leonard|last2=Jerison|first2=Meyer|title=Rings of continuous functions|publisher=Springer-Verlag|year=1976}}.
| |
| *{{citation | last=Kelley |first=John |title=General topology |publisher=Springer-Verlag |year=1955 |series=Graduate Texts in Mathematics |volume=27}}.
| |
| *{{Citation | last1=Kline | first1=Morris | author1-link=Morris Kline | title=Mathematical thought from ancient to modern times | year=1972 | publisher=[[Oxford University Press]] | edition=3rd | isbn=978-0-19-506136-9 | publication-date=1990}}.
| |
| *{{citation |first=Henri |last=Lebesgue |title=Leçons sur l'intégration et la recherche des fonctions primitives |url=http://books.google.com/?id=VfUKAAAAYAAJ&dq=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1#v=onepage&q= |year=1904 |publisher=Gauthier-Villars}}.
| |
| *{{Citation | last1=Robinson | first1=Abraham | author1-link=Abraham Robinson | title=Non-standard analysis | publisher=[[Princeton University Press]] | isbn=978-0-691-04490-3 | id={{MathSciNet | id = 0205854}} | year=1996}}.
| |
| *{{citation |first=C.T. |last= Scarborough |first2= A.H. |last2= Stone |title= Products of nearly compact spaces |journal= Transactions of the American Mathematical Society |volume= 124 |year=1966 |pages= 131–147 |doi=10.2307/1994440 |issue=1 |publisher=Transactions of the American Mathematical Society, Vol. 124, No. 1 |jstor=1994440}}.
| |
| * {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
| |
| | |
| ==External links==
| |
| * {{planetmathref|id=1233|title=Countably compact}}
| |
| * {{cite arXiv |last=Sundström |first=Manya Raman | eprint=1006.4131 |title=A pedagogical history of compactness |class=math.HO |year=2010 |version=v1}}
| |
| ----
| |
| {{PlanetMath attribution|id=3133|title=Examples of compact spaces}}
| |
| | |
| {{DEFAULTSORT:Compact Space}}
| |
| [[Category:Compactness (mathematics)]]
| |
| [[Category:General topology]]
| |
| [[Category:Properties of topological spaces]]
| |
| [[Category:Topology]]
| |