{"history":[],"article":{"id":"143d1494-796f-4f81-ac65-794f14156eeb","title":"Compact space","slug":"compact-space","content":":::summary\n:::\n\n# \nCompact.svg, the interval\nis not compact because it is not bounded.\nThe interval\nis not compact because it is not closed (but bounded).\nThe interval\nis compact because it is both closed and bounded.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), specifically [general topology](https://bliptext.com/articles/general-topology), **compactness** is a property that seeks to generalize the notion of a [closed](https://bliptext.com/articles/closed-set) and [bounded](https://bliptext.com/articles/bounded-set) subset of [Euclidean space](https://bliptext.com/articles/euclidean-space).\nThe idea is that a compact space has no \"punctures\" or \"missing endpoints\", i.e., it includes all limiting values of points.\nFor example, the open [interval](https://bliptext.com/articles/interval-mathematics) (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed [interval](https://bliptext.com/articles/interval-mathematics) [0,1] would be compact.\nSimilarly, the space of [rational numbers](https://bliptext.com/articles/rational-number) \\mathbb{Q} is not compact, because it has infinitely many \"punctures\" corresponding to the ir[rational numbers](https://bliptext.com/articles/rational-number), and the space of [real numbers](https://bliptext.com/articles/real-number) \\mathbb{R} is not compact either, because it excludes the two limiting values +\\infty and -\\infty.\nHowever, the extended real number line would be compact, since it contains both infinities.\nThere are many ways to make this heuristic notion precise.\nThese ways usually agree in a [metric space](https://bliptext.com/articles/metric-space), but may not be [equivalent](https://bliptext.com/articles/logical-equivalence) in other [topological spaces](https://bliptext.com/articles/topological-space).\nOne such generalization is that a topological space is sequentially compact if every [infinite sequence](https://bliptext.com/articles/infinite-sequence) of points sampled from the space has an infinite [subsequence](https://bliptext.com/articles/subsequence) that converges to some point of the space.\nThe [Bolzano–Weierstrass theorem](https://bliptext.com/articles/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.\nThus, if one chooses an infinite number of points in the closed [unit interval](https://bliptext.com/articles/unit-interval)\n[0, 1]\n, some of those points will get arbitrarily close to some real number in that space.\nFor instance, some of the numbers in the sequence 1⁄2, 4⁄5, 1⁄3, 5⁄6, 1⁄4, 6⁄7, ... accumulate to 0 (while others accumulate to 1).\nSince neither 0 nor 1 are members of the open unit interval\n(0, 1)\n, those same sets of points would not accumulate to any point of it, so the open unit interval is not compact.\nAlthough subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded.\nFor example, considering (the real number line), the sequence of points 0, 1, 2, 3, ... has no subsequence that converges to any real number.\nCompactness was formally introduced by [Maurice Fréchet](https://bliptext.com/articles/maurice-fr-chet) in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to [spaces of functions](https://bliptext.com/articles/function-space).\nThe [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem) and the [Peano existence theorem](https://bliptext.com/articles/peano-existence-theorem) exemplify applications of this notion of compactness to classical analysis.\nFollowing its initial introduction, various equivalent notions of compactness, including [sequential compactness](https://bliptext.com/articles/sequentially-compact-space) and [limit point compactness](https://bliptext.com/articles/limit-point-compact), were developed in general [metric spaces](https://bliptext.com/articles/metric-space).\nIn general topological spaces, however, these notions of compactness are not necessarily equivalent.\nThe most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of [open sets](https://bliptext.com/articles/open-set) that \"[cover](https://bliptext.com/articles/cover-topology)\" the space, in the sense that each point of the space lies in some set contained in the family.\nThis more subtle notion, introduced by [Pavel Alexandrov](https://bliptext.com/articles/pavel-alexandrov) and [Pavel Urysohn](https://bliptext.com/articles/pavel-urysohn) in 1929, exhibits compact spaces as generalizations of [finite sets](https://bliptext.com/articles/finite-set).\nIn spaces that are compact in this sense, it is often possible to patch together information that holds [locally](https://bliptext.com/articles/local-property) – that is, in a neighborhood of each point – into corresponding statements that hold throughout the space, and many theorems are of this character.\nThe term **compact set** is sometimes used as a synonym for compact space, but also often refers to a [compact subspace](https://bliptext.com/articles/) of a [topological space](https://bliptext.com/articles/topological-space).\n\n# Historical development\nIn the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness.\nOn the one hand, [Bernard Bolzano](https://bliptext.com/articles/bernard-bolzano) ([1817](https://bliptext.com/articles/)) 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](https://bliptext.com/articles/limit-point-of-a-sequence).\nBolzano's proof relied on the [method of bisection](https://bliptext.com/articles/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.\nThe 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.\nThe full significance of [Bolzano's theorem](https://bliptext.com/articles/bolzano-weierstrass-theorem), and its method of proof, would not emerge until almost 50 years later when it was rediscovered by [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass).\nIn the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for [spaces of functions](https://bliptext.com/articles/function-space) rather than just numbers or geometrical points.\nThe idea of regarding functions as themselves points of a generalized space dates back to the investigations of [Giulio Ascoli](https://bliptext.com/articles/giulio-ascoli) and [Cesare Arzelà](https://bliptext.com/articles/cesare-arzel).\nThe culmination of their investigations, the [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), was a generalization of the Bolzano–Weierstrass theorem to families of [continuous functions](https://bliptext.com/articles/continuous-function), the precise conclusion of which was that it was possible to extract a [uniformly convergent](https://bliptext.com/articles/uniform-convergence) sequence of functions from a suitable family of functions.\nThe uniform limit of this sequence then played precisely the same role as Bolzano's \"limit point\".\nTowards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of [integral equations](https://bliptext.com/articles/integral-equation), as investigated by [David Hilbert](https://bliptext.com/articles/david-hilbert) and [Erhard Schmidt](https://bliptext.com/articles/erhard-schmidt).\nFor a certain class of [Green's functions](https://bliptext.com/articles/green-s-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](https://bliptext.com/articles/mean-convergence) – or convergence in what would later be dubbed a [Hilbert space](https://bliptext.com/articles/hilbert-space).\nThis ultimately led to the notion of a [compact operator](https://bliptext.com/articles/compact-operator) as an offshoot of the general notion of a compact space.\nIt was [Maurice Fréchet](https://bliptext.com/articles/maurice-ren-fr-chet) who, in [1906](https://bliptext.com/articles/), had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paper which led to the famous [1906](https://bliptext.com/articles/) thesis).\nHowever, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the [continuum](https://bliptext.com/articles/linear-continuum), which was seen as fundamental for the rigorous formulation of analysis.\nIn 1870, [Eduard Heine](https://bliptext.com/articles/eduard-heine) showed that a [continuous function](https://bliptext.com/articles/continuous-function) defined on a closed and bounded interval was in fact [uniformly continuous](https://bliptext.com/articles/uniformly-continuous).\nIn 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.\nThe significance of this lemma was recognized by [Émile Borel](https://bliptext.com/articles/mile-borel) ([1895](https://bliptext.com/articles/)), and it was generalized to arbitrary collections of intervals by [Pierre Cousin](https://bliptext.com/articles/pierre-cousin-mathematician) ([1895](https://bliptext.com/articles/)) and [Henri Lebesgue](https://bliptext.com/articles/henri-lebesgue) ([1904](https://bliptext.com/articles/)).\nThe [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem), as the result is now known, is another special property possessed by closed and bounded sets of real numbers.\nThis property was significant because it allowed for the passage from [local information](https://bliptext.com/articles/local-property) about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function).\nThis sentiment was expressed by, who also exploited it in the development of the [integral now bearing his name](https://bliptext.com/articles/lebesgue-integral).\nUltimately, the Russian school of [point-set topology](https://bliptext.com/articles/point-set-topology), under the direction of [Pavel Alexandrov](https://bliptext.com/articles/pavel-alexandrov) and [Pavel Urysohn](https://bliptext.com/articles/pavel-urysohn), formulated Heine–Borel compactness in a way that could be applied to the modern notion of a [topological space](https://bliptext.com/articles/topological-space).\nshowed that the earlier version of compactness due to Fréchet, now called (relative) [sequential compactness](https://bliptext.com/articles/sequential-compactness), under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers.\nIt 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.\n\n# Basic examples\nAny [finite space](https://bliptext.com/articles/finite-topological-space) is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it.\nA nontrivial example of a compact space is the (closed) [unit interval](https://bliptext.com/articles/unit-interval) of [real numbers](https://bliptext.com/articles/real-number).\nIf one chooses an infinite number of distinct points in the unit interval, then there must be some [accumulation point](https://bliptext.com/articles/accumulation-point) among these points in that interval.\nFor instance, the odd-numbered terms of the sequence 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.\nThe given example sequence shows the importance of including the [boundary](https://bliptext.com/articles/boundary-topology) points of the interval, since the [limit points](https://bliptext.com/articles/limit-of-a-sequence) must be in the space itself — an open (or half-open) interval of the real numbers is not compact.\nIt is also crucial that the interval be [bounded](https://bliptext.com/articles/bounded-set), since in the interval, one could choose the sequence of points 0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number.\nIn two dimensions, closed [disks](https://bliptext.com/articles/disk-mathematics) 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.\nHowever, 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.\nLikewise, 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.\nLines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.\n\n# Definitions\nVarious definitions of compactness may apply, depending on the level of generality.\nA subset of [Euclidean space](https://bliptext.com/articles/euclidean-space) in particular is called compact if it is [closed](https://bliptext.com/articles/closed-set) and [bounded](https://bliptext.com/articles/bounded-set).\nThis implies, by the [Bolzano–Weierstrass theorem](https://bliptext.com/articles/bolzano-weierstrass-theorem), that any infinite [sequence](https://bliptext.com/articles/sequence-mathematics) from the set has a sub[sequence](https://bliptext.com/articles/sequence-mathematics) that converges to a point in the set.\nVarious equivalent notions of compactness, such as [sequential compactness](https://bliptext.com/articles/sequential-compactness) and [limit point compactness](https://bliptext.com/articles/limit-point-compact), can be developed in general [metric spaces](https://bliptext.com/articles/metric-space).\nIn contrast, the different notions of compactness are not equivalent in general [topological spaces](https://bliptext.com/articles/topological-space), and the most useful notion of compactness – originally called bicompactness – is defined using [covers](https://bliptext.com/articles/cover-topology) consisting of [open sets](https://bliptext.com/articles/open-set) (see Open cover definition below).\nThat this form of compactness holds for closed and bounded subsets of Euclidean space is known as the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nCompactness, when defined in this manner, often allows one to take information that is known [locally](https://bliptext.com/articles/local-property) – in a [neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics) of each point of the space – and to extend it to information that holds globally throughout the space.\nAn 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](https://bliptext.com/articles/uniformly-continuous); here, continuity is a local property of the function, and uniform continuity the corresponding global property.\n\n## Open cover definition\nFormally, a [topological space](https://bliptext.com/articles/topological-space) X is called compact if every [open cover](https://bliptext.com/articles/open-cover) of X has a [finite](https://bliptext.com/articles/finite-set) [subcover](https://bliptext.com/articles/subcover).\nThat is, X is compact if for every collection C of open subsets of X such that\nthere is a **finite** subcollection F ⊆ C such that\nSome branches of mathematics such as [algebraic geometry](https://bliptext.com/articles/algebraic-geometry), typically influenced by the French school of [Bourbaki](https://bliptext.com/articles/nicolas-bourbaki), use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both [Hausdorff](https://bliptext.com/articles/hausdorff-space) and quasi-compact.\nA compact set is sometimes referred to as a compactum, plural compacta.\n\n## Compactness of subsets\nA subset K of a topological space X is said to be compact if it is compact as a subspace (in the [subspace topology](https://bliptext.com/articles/subspace-topology)).\nThat is, K is compact if for every arbitrary collection C of open subsets of X such that\nthere is a **finite** subcollection F ⊆ C such that\nBecause compactness is a [topological property](https://bliptext.com/articles/topological-property), the compactness of a subset depends only on the subspace topology induced on it.\nIt follows that, if, with subset Z equipped with the subspace topology, then K is compact in Z if and only if K is compact in Y.\n\n## Characterization\nIf X is a topological space then the following are equivalent:\nBourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).\n\n### Euclidean space\nFor any [subset](https://bliptext.com/articles/subset) A of [Euclidean space](https://bliptext.com/articles/euclidean-space), A is compact if and only if it is [closed](https://bliptext.com/articles/closed-set) and [bounded](https://bliptext.com/articles/bounded-set); this is the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nAs a [Euclidean space](https://bliptext.com/articles/euclidean-space) is a metric space, the conditions in the next subsection also apply to all of its subsets.\nOf 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](https://bliptext.com/articles/interval-mathematics) or closed n-ball.\n\n### Metric spaces\nFor any metric space\n(X, d)\n, the following are equivalent (assuming [countable choice](https://bliptext.com/articles/countable-choice)):\n(X, d)\nis compact.\n(X, d)\nis [complete](https://bliptext.com/articles/completeness-topology) and [totally bounded](https://bliptext.com/articles/totally-bounded) (this is also equivalent to compactness for [uniform spaces](https://bliptext.com/articles/uniform-space)).\n(X, d)\nis sequentially compact; that is, every [sequence](https://bliptext.com/articles/sequence) in X has a convergent sub[sequence](https://bliptext.com/articles/sequence) whose limit is in X (this is also equivalent to compactness for [first-countable](https://bliptext.com/articles/first-countable) [uniform spaces](https://bliptext.com/articles/uniform-space)).\n(X, d)\nis [limit point compact](https://bliptext.com/articles/limit-point-compact) (also called weakly countably compact); that is, every infinite subset of X has at least one [limit point](https://bliptext.com/articles/limit-point-of-a-set) in X.\n(X, d)\nis [countably compact](https://bliptext.com/articles/countably-compact); that is, every countable open cover of X has a finite subcover.\n(X, d)\nis an image of a continuous function from the [Cantor set](https://bliptext.com/articles/cantor-set).\nS1 ⊇ S2 ⊇ ...\nin\n(X, d)\nhas a nonempty intersection.\nS1 ⊆ S2 ⊆ ...\nin\n(X, d)\nfails to cover X.\nA compact metric space\n(X, d)\nalso satisfies the following properties:\n(X, d)\nis [second-countable](https://bliptext.com/articles/second-countable-space), [separable](https://bliptext.com/articles/separable-space) and [Lindelöf](https://bliptext.com/articles/lindel-f-space) – these three conditions are equivalent for metric spaces.\nThe converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.\n\n### Ordered spaces\nFor an ordered space\n(X, <)\n(i.e. a totally ordered set equipped with the order topology), the following are equivalent:\n(X, <)\nis compact.\nAn ordered space satisfying (any one of) these conditions is called a complete lattice.\nIn addition, the following are equivalent for all ordered spaces\n(X, <)\n, and (assuming [countable choice](https://bliptext.com/articles/countable-choice)) are true whenever\n(X, <)\nis compact.\n(The converse in general fails if\n(X, <)\nis not also metrizable.):\n(X, <)\nhas a subsequence that converges in\n(X, <)\n.\n(X, <)\nhas a nonempty intersection.\n(X, <)\nfails to cover X.\n\n### Characterization by continuous functions\nLet X be a topological space and\nC(X)\nthe ring of real continuous functions on X. For each\np ∈ X\n, the evaluation map\ngiven by\nevp(f) = f(p)\nis a ring homomorphism.\nThe [kernel](https://bliptext.com/articles/kernel-algebra) of\nevp\nis a [maximal ideal](https://bliptext.com/articles/maximal-ideal), since the [residue field](https://bliptext.com/articles/residue-field) C(X)/ker evp is the field of real numbers, by the [first isomorphism theorem](https://bliptext.com/articles/first-isomorphism-theorem).\nA topological space X is [pseudocompact](https://bliptext.com/articles/pseudocompact-space) if and only if every maximal ideal in\nC(X)\nhas residue field the real numbers.\nFor [completely regular spaces](https://bliptext.com/articles/completely-regular-space), this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.\nThere are pseudocompact spaces that are not compact, though.\nIn general, for non-pseudocompact spaces there are always maximal ideals m in\nC(X)\nsuch that the residue field\nC(X)/m\nis a ([non-Archimedean](https://bliptext.com/articles/non-archimedean-field)) [hyperreal field](https://bliptext.com/articles/hyperreal-field).\nThe framework of [non-standard analysis](https://bliptext.com/articles/non-standard-analysis) allows for the following alternative characterization of compactness: a topological space X is compact if and only if every point x of the natural extension\nis [infinitely close](https://bliptext.com/articles/infinitesimal) to a point\nx0\nof X (more precisely, x is contained in the [monad](https://bliptext.com/articles/monad-non-standard-analysis) of\nx0\n).\n\n### Hyperreal definition\nA space X is compact if its [hyperreal extension](https://bliptext.com/articles/hyperreal-number)\n(constructed, for example, by the [ultrapower construction](https://bliptext.com/articles/ultrapower-construction)) has the property that every point of\nis infinitely close to some point of\nX ⊂ *X\n. For example, an open real interval is not compact because its hyperreal extension\ncontains infinitesimals, which are infinitely close to 0, which is not a point of X.\n\n# Sufficient conditions\n\n# Properties of compact spaces\nA ⊆ U\nand\nB ⊆ V\n.\n(X, d)\nis compact then it is d-bounded.\n\n## Functions and compact spaces\nSince a [continuous](https://bliptext.com/articles/continuous-function-topology) image of a compact space is compact, the [extreme value theorem](https://bliptext.com/articles/extreme-value-theorem) holds for such spaces: a [continuous](https://bliptext.com/articles/continuous-function-topology) real-valued function on a nonempty compact space is bounded above and attains its supremum.\n(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](https://bliptext.com/articles/proper-map) is compact.\n\n## Compactifications\nEvery topological space X is an open [dense subspace](https://bliptext.com/articles/dense-topological-subspace) of a compact space having at most one point more than X, by the [Alexandroff one-point compactification](https://bliptext.com/articles/alexandroff-one-point-compactification).\nBy the same construction, every [locally compact](https://bliptext.com/articles/locally-compact) Hausdorff space X is an open dense subspace of a compact Hausdorff space having at most one point more than X.\n\n## Ordered compact spaces\nA nonempty compact subset of the [real numbers](https://bliptext.com/articles/real-number) has a greatest element and a least element.\nLet X be a [simply ordered](https://bliptext.com/articles/total-order) set endowed with the [order topology](https://bliptext.com/articles/order-topology).\nThen X is compact if and only if X is a [complete lattice](https://bliptext.com/articles/complete-lattice) (i.e. all subsets have suprema and infima).\n\n# Examples\nS1\nS2\n. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.\nis compact.\nThis follows from the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nThe open interval is not compact: the [open cover](https://bliptext.com/articles/open-cover) for\nn = 3, 4, ...\ndoes not have a finite subcover.\nSimilarly, the set of [rational numbers](https://bliptext.com/articles/rational-numbers) in the closed interval is not compact: the sets of [rational numbers](https://bliptext.com/articles/rational-numbers) in the intervals cover all the rationals in [0, 1] for\nn = 4, 5, ...\nbut this cover does not have a finite subcover.\nHere, the sets are open in the subspace topology even though they are not open as subsets of \\mathbb{R}.\n, where n takes all integer values in\nZ\n, cover \\mathbb{R} but there is no finite subcover.\nf : \\mathbb{R} → [0, 1]\nfrom the real number line to the closed unit interval, and define a topology on K so that a sequence \\{f_n\\} in K converges towards\nf ∈ K\nif and only if \\{f_n(x)\\} converges towards\nf(x)\nfor all real numbers x. There is only one such topology; it is called the topology of [pointwise convergence](https://bliptext.com/articles/pointwise-convergence) or the [product topology](https://bliptext.com/articles/product-topology).\nThen K is a compact topological space; this follows from the [Tychonoff theorem](https://bliptext.com/articles/tychonoff-theorem).\nf : →\nsatisfying the [Lipschitz condition](https://bliptext.com/articles/lipschitz-condition)\n≤\nfor all\nx, y ∈\n. Consider on K the metric induced by the [uniform distance](https://bliptext.com/articles/uniform-convergence) Then by the Arzelà–Ascoli theorem the space K is compact.\n\n## Algebraic examples","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:30:03.056Z","updated_at":"2025-01-25T16:30:03.056Z","search_vector":"'compact':1 'space':2","search_content":" \n \n\n# \nCompact.svg, the interval\nis not compact because it is not bounded.\nThe interval\nis not compact because it is not closed (but bounded).\nThe interval\nis compact because it is both closed and bounded.]]\nIn [mathematics](https://bliptext.com/articles/mathematics), specifically [general topology](https://bliptext.com/articles/general-topology), **compactness** is a property that seeks to generalize the notion of a [closed](https://bliptext.com/articles/closed-set) and [bounded](https://bliptext.com/articles/bounded-set) subset of [Euclidean space](https://bliptext.com/articles/euclidean-space).\nThe idea is that a compact space has no \"punctures\" or \"missing endpoints\", i.e., it includes all limiting values of points.\nFor example, the open [interval](https://bliptext.com/articles/interval-mathematics) (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed [interval](https://bliptext.com/articles/interval-mathematics) [0,1] would be compact.\nSimilarly, the space of [rational numbers](https://bliptext.com/articles/rational-number) \\mathbb{Q} is not compact, because it has infinitely many \"punctures\" corresponding to the ir[rational numbers](https://bliptext.com/articles/rational-number), and the space of [real numbers](https://bliptext.com/articles/real-number) \\mathbb{R} is not compact either, because it excludes the two limiting values +\\infty and -\\infty.\nHowever, the extended real number line would be compact, since it contains both infinities.\nThere are many ways to make this heuristic notion precise.\nThese ways usually agree in a [metric space](https://bliptext.com/articles/metric-space), but may not be [equivalent](https://bliptext.com/articles/logical-equivalence) in other [topological spaces](https://bliptext.com/articles/topological-space).\nOne such generalization is that a topological space is sequentially compact if every [infinite sequence](https://bliptext.com/articles/infinite-sequence) of points sampled from the space has an infinite [subsequence](https://bliptext.com/articles/subsequence) that converges to some point of the space.\nThe [Bolzano–Weierstrass theorem](https://bliptext.com/articles/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.\nThus, if one chooses an infinite number of points in the closed [unit interval](https://bliptext.com/articles/unit-interval)\n[0, 1]\n, some of those points will get arbitrarily close to some real number in that space.\nFor instance, some of the numbers in the sequence 1⁄2, 4⁄5, 1⁄3, 5⁄6, 1⁄4, 6⁄7, ... accumulate to 0 (while others accumulate to 1).\nSince neither 0 nor 1 are members of the open unit interval\n(0, 1)\n, those same sets of points would not accumulate to any point of it, so the open unit interval is not compact.\nAlthough subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded.\nFor example, considering (the real number line), the sequence of points 0, 1, 2, 3, ... has no subsequence that converges to any real number.\nCompactness was formally introduced by [Maurice Fréchet](https://bliptext.com/articles/maurice-fr-chet) in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to [spaces of functions](https://bliptext.com/articles/function-space).\nThe [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem) and the [Peano existence theorem](https://bliptext.com/articles/peano-existence-theorem) exemplify applications of this notion of compactness to classical analysis.\nFollowing its initial introduction, various equivalent notions of compactness, including [sequential compactness](https://bliptext.com/articles/sequentially-compact-space) and [limit point compactness](https://bliptext.com/articles/limit-point-compact), were developed in general [metric spaces](https://bliptext.com/articles/metric-space).\nIn general topological spaces, however, these notions of compactness are not necessarily equivalent.\nThe most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of [open sets](https://bliptext.com/articles/open-set) that \"[cover](https://bliptext.com/articles/cover-topology)\" the space, in the sense that each point of the space lies in some set contained in the family.\nThis more subtle notion, introduced by [Pavel Alexandrov](https://bliptext.com/articles/pavel-alexandrov) and [Pavel Urysohn](https://bliptext.com/articles/pavel-urysohn) in 1929, exhibits compact spaces as generalizations of [finite sets](https://bliptext.com/articles/finite-set).\nIn spaces that are compact in this sense, it is often possible to patch together information that holds [locally](https://bliptext.com/articles/local-property) – that is, in a neighborhood of each point – into corresponding statements that hold throughout the space, and many theorems are of this character.\nThe term **compact set** is sometimes used as a synonym for compact space, but also often refers to a [compact subspace](https://bliptext.com/articles/) of a [topological space](https://bliptext.com/articles/topological-space).\n\n# Historical development\nIn the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness.\nOn the one hand, [Bernard Bolzano](https://bliptext.com/articles/bernard-bolzano) ([1817](https://bliptext.com/articles/)) 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](https://bliptext.com/articles/limit-point-of-a-sequence).\nBolzano's proof relied on the [method of bisection](https://bliptext.com/articles/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.\nThe 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.\nThe full significance of [Bolzano's theorem](https://bliptext.com/articles/bolzano-weierstrass-theorem), and its method of proof, would not emerge until almost 50 years later when it was rediscovered by [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass).\nIn the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for [spaces of functions](https://bliptext.com/articles/function-space) rather than just numbers or geometrical points.\nThe idea of regarding functions as themselves points of a generalized space dates back to the investigations of [Giulio Ascoli](https://bliptext.com/articles/giulio-ascoli) and [Cesare Arzelà](https://bliptext.com/articles/cesare-arzel).\nThe culmination of their investigations, the [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem), was a generalization of the Bolzano–Weierstrass theorem to families of [continuous functions](https://bliptext.com/articles/continuous-function), the precise conclusion of which was that it was possible to extract a [uniformly convergent](https://bliptext.com/articles/uniform-convergence) sequence of functions from a suitable family of functions.\nThe uniform limit of this sequence then played precisely the same role as Bolzano's \"limit point\".\nTowards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of [integral equations](https://bliptext.com/articles/integral-equation), as investigated by [David Hilbert](https://bliptext.com/articles/david-hilbert) and [Erhard Schmidt](https://bliptext.com/articles/erhard-schmidt).\nFor a certain class of [Green's functions](https://bliptext.com/articles/green-s-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](https://bliptext.com/articles/mean-convergence) – or convergence in what would later be dubbed a [Hilbert space](https://bliptext.com/articles/hilbert-space).\nThis ultimately led to the notion of a [compact operator](https://bliptext.com/articles/compact-operator) as an offshoot of the general notion of a compact space.\nIt was [Maurice Fréchet](https://bliptext.com/articles/maurice-ren-fr-chet) who, in [1906](https://bliptext.com/articles/), had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paper which led to the famous [1906](https://bliptext.com/articles/) thesis).\nHowever, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the [continuum](https://bliptext.com/articles/linear-continuum), which was seen as fundamental for the rigorous formulation of analysis.\nIn 1870, [Eduard Heine](https://bliptext.com/articles/eduard-heine) showed that a [continuous function](https://bliptext.com/articles/continuous-function) defined on a closed and bounded interval was in fact [uniformly continuous](https://bliptext.com/articles/uniformly-continuous).\nIn 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.\nThe significance of this lemma was recognized by [Émile Borel](https://bliptext.com/articles/mile-borel) ([1895](https://bliptext.com/articles/)), and it was generalized to arbitrary collections of intervals by [Pierre Cousin](https://bliptext.com/articles/pierre-cousin-mathematician) ([1895](https://bliptext.com/articles/)) and [Henri Lebesgue](https://bliptext.com/articles/henri-lebesgue) ([1904](https://bliptext.com/articles/)).\nThe [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem), as the result is now known, is another special property possessed by closed and bounded sets of real numbers.\nThis property was significant because it allowed for the passage from [local information](https://bliptext.com/articles/local-property) about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function).\nThis sentiment was expressed by, who also exploited it in the development of the [integral now bearing his name](https://bliptext.com/articles/lebesgue-integral).\nUltimately, the Russian school of [point-set topology](https://bliptext.com/articles/point-set-topology), under the direction of [Pavel Alexandrov](https://bliptext.com/articles/pavel-alexandrov) and [Pavel Urysohn](https://bliptext.com/articles/pavel-urysohn), formulated Heine–Borel compactness in a way that could be applied to the modern notion of a [topological space](https://bliptext.com/articles/topological-space).\nshowed that the earlier version of compactness due to Fréchet, now called (relative) [sequential compactness](https://bliptext.com/articles/sequential-compactness), under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers.\nIt 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.\n\n# Basic examples\nAny [finite space](https://bliptext.com/articles/finite-topological-space) is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it.\nA nontrivial example of a compact space is the (closed) [unit interval](https://bliptext.com/articles/unit-interval) of [real numbers](https://bliptext.com/articles/real-number).\nIf one chooses an infinite number of distinct points in the unit interval, then there must be some [accumulation point](https://bliptext.com/articles/accumulation-point) among these points in that interval.\nFor instance, the odd-numbered terms of the sequence 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.\nThe given example sequence shows the importance of including the [boundary](https://bliptext.com/articles/boundary-topology) points of the interval, since the [limit points](https://bliptext.com/articles/limit-of-a-sequence) must be in the space itself — an open (or half-open) interval of the real numbers is not compact.\nIt is also crucial that the interval be [bounded](https://bliptext.com/articles/bounded-set), since in the interval, one could choose the sequence of points 0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number.\nIn two dimensions, closed [disks](https://bliptext.com/articles/disk-mathematics) 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.\nHowever, 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.\nLikewise, 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.\nLines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.\n\n# Definitions\nVarious definitions of compactness may apply, depending on the level of generality.\nA subset of [Euclidean space](https://bliptext.com/articles/euclidean-space) in particular is called compact if it is [closed](https://bliptext.com/articles/closed-set) and [bounded](https://bliptext.com/articles/bounded-set).\nThis implies, by the [Bolzano–Weierstrass theorem](https://bliptext.com/articles/bolzano-weierstrass-theorem), that any infinite [sequence](https://bliptext.com/articles/sequence-mathematics) from the set has a sub[sequence](https://bliptext.com/articles/sequence-mathematics) that converges to a point in the set.\nVarious equivalent notions of compactness, such as [sequential compactness](https://bliptext.com/articles/sequential-compactness) and [limit point compactness](https://bliptext.com/articles/limit-point-compact), can be developed in general [metric spaces](https://bliptext.com/articles/metric-space).\nIn contrast, the different notions of compactness are not equivalent in general [topological spaces](https://bliptext.com/articles/topological-space), and the most useful notion of compactness – originally called bicompactness – is defined using [covers](https://bliptext.com/articles/cover-topology) consisting of [open sets](https://bliptext.com/articles/open-set) (see Open cover definition below).\nThat this form of compactness holds for closed and bounded subsets of Euclidean space is known as the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nCompactness, when defined in this manner, often allows one to take information that is known [locally](https://bliptext.com/articles/local-property) – in a [neighbourhood](https://bliptext.com/articles/neighbourhood-mathematics) of each point of the space – and to extend it to information that holds globally throughout the space.\nAn 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](https://bliptext.com/articles/uniformly-continuous); here, continuity is a local property of the function, and uniform continuity the corresponding global property.\n\n## Open cover definition\nFormally, a [topological space](https://bliptext.com/articles/topological-space) X is called compact if every [open cover](https://bliptext.com/articles/open-cover) of X has a [finite](https://bliptext.com/articles/finite-set) [subcover](https://bliptext.com/articles/subcover).\nThat is, X is compact if for every collection C of open subsets of X such that\nthere is a **finite** subcollection F ⊆ C such that\nSome branches of mathematics such as [algebraic geometry](https://bliptext.com/articles/algebraic-geometry), typically influenced by the French school of [Bourbaki](https://bliptext.com/articles/nicolas-bourbaki), use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both [Hausdorff](https://bliptext.com/articles/hausdorff-space) and quasi-compact.\nA compact set is sometimes referred to as a compactum, plural compacta.\n\n## Compactness of subsets\nA subset K of a topological space X is said to be compact if it is compact as a subspace (in the [subspace topology](https://bliptext.com/articles/subspace-topology)).\nThat is, K is compact if for every arbitrary collection C of open subsets of X such that\nthere is a **finite** subcollection F ⊆ C such that\nBecause compactness is a [topological property](https://bliptext.com/articles/topological-property), the compactness of a subset depends only on the subspace topology induced on it.\nIt follows that, if, with subset Z equipped with the subspace topology, then K is compact in Z if and only if K is compact in Y.\n\n## Characterization\nIf X is a topological space then the following are equivalent:\nBourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).\n\n### Euclidean space\nFor any [subset](https://bliptext.com/articles/subset) A of [Euclidean space](https://bliptext.com/articles/euclidean-space), A is compact if and only if it is [closed](https://bliptext.com/articles/closed-set) and [bounded](https://bliptext.com/articles/bounded-set); this is the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nAs a [Euclidean space](https://bliptext.com/articles/euclidean-space) is a metric space, the conditions in the next subsection also apply to all of its subsets.\nOf 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](https://bliptext.com/articles/interval-mathematics) or closed n-ball.\n\n### Metric spaces\nFor any metric space\n(X, d)\n, the following are equivalent (assuming [countable choice](https://bliptext.com/articles/countable-choice)):\n(X, d)\nis compact.\n(X, d)\nis [complete](https://bliptext.com/articles/completeness-topology) and [totally bounded](https://bliptext.com/articles/totally-bounded) (this is also equivalent to compactness for [uniform spaces](https://bliptext.com/articles/uniform-space)).\n(X, d)\nis sequentially compact; that is, every [sequence](https://bliptext.com/articles/sequence) in X has a convergent sub[sequence](https://bliptext.com/articles/sequence) whose limit is in X (this is also equivalent to compactness for [first-countable](https://bliptext.com/articles/first-countable) [uniform spaces](https://bliptext.com/articles/uniform-space)).\n(X, d)\nis [[limit point](https://bliptext.com/articles/limit-point-of-a-set) compact](https://bliptext.com/articles/limit-point-compact) (also called weakly countably compact); that is, every infinite subset of X has at least one [limit point](https://bliptext.com/articles/limit-point-of-a-set) in X.\n(X, d)\nis [countably compact](https://bliptext.com/articles/countably-compact); that is, every countable open cover of X has a finite subcover.\n(X, d)\nis an image of a continuous function from the [Cantor set](https://bliptext.com/articles/cantor-set).\nS1 ⊇ S2 ⊇ ...\nin\n(X, d)\nhas a nonempty intersection.\nS1 ⊆ S2 ⊆ ...\nin\n(X, d)\nfails to cover X.\nA compact metric space\n(X, d)\nalso satisfies the following properties:\n(X, d)\nis [second-countable](https://bliptext.com/articles/second-countable-space), [separable](https://bliptext.com/articles/separable-space) and [Lindelöf](https://bliptext.com/articles/lindel-f-space) – these three conditions are equivalent for metric spaces.\nThe converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.\n\n### Ordered spaces\nFor an ordered space\n(X, <)\n(i.e. a totally ordered set equipped with the order topology), the following are equivalent:\n(X, <)\nis compact.\nAn ordered space satisfying (any one of) these conditions is called a complete lattice.\nIn addition, the following are equivalent for all ordered spaces\n(X, <)\n, and (assuming [countable choice](https://bliptext.com/articles/countable-choice)) are true whenever\n(X, <)\nis compact.\n(The converse in general fails if\n(X, <)\nis not also metrizable.):\n(X, <)\nhas a subsequence that converges in\n(X, <)\n.\n(X, <)\nhas a nonempty intersection.\n(X, <)\nfails to cover X.\n\n### Characterization by continuous functions\nLet X be a topological space and\nC(X)\nthe ring of real continuous functions on X. For each\np ∈ X\n, the evaluation map\ngiven by\nevp(f) = f(p)\nis a ring homomorphism.\nThe [kernel](https://bliptext.com/articles/kernel-algebra) of\nevp\nis a [maximal ideal](https://bliptext.com/articles/maximal-ideal), since the [residue field](https://bliptext.com/articles/residue-field) C(X)/ker evp is the field of real numbers, by the [first isomorphism theorem](https://bliptext.com/articles/first-isomorphism-theorem).\nA topological space X is [pseudocompact](https://bliptext.com/articles/pseudocompact-space) if and only if every maximal ideal in\nC(X)\nhas residue field the real numbers.\nFor [completely regular spaces](https://bliptext.com/articles/completely-regular-space), this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.\nThere are pseudocompact spaces that are not compact, though.\nIn general, for non-pseudocompact spaces there are always maximal ideals m in\nC(X)\nsuch that the residue field\nC(X)/m\nis a ([non-Archimedean](https://bliptext.com/articles/non-archimedean-field)) [hyperreal field](https://bliptext.com/articles/hyperreal-field).\nThe framework of [non-standard analysis](https://bliptext.com/articles/non-standard-analysis) allows for the following alternative characterization of compactness: a topological space X is compact if and only if every point x of the natural extension\nis [infinitely close](https://bliptext.com/articles/infinitesimal) to a point\nx0\nof X (more precisely, x is contained in the [monad](https://bliptext.com/articles/monad-non-standard-analysis) of\nx0\n).\n\n### Hyperreal definition\nA space X is compact if its [hyperreal extension](https://bliptext.com/articles/hyperreal-number)\n(constructed, for example, by the [ultrapower construction](https://bliptext.com/articles/ultrapower-construction)) has the property that every point of\nis infinitely close to some point of\nX ⊂ *X\n. For example, an open real interval is not compact because its hyperreal extension\ncontains infinitesimals, which are infinitely close to 0, which is not a point of X.\n\n# Sufficient conditions\n\n# Properties of compact spaces\nA ⊆ U\nand\nB ⊆ V\n.\n(X, d)\nis compact then it is d-bounded.\n\n## Functions and compact spaces\nSince a [continuous](https://bliptext.com/articles/continuous-function-topology) image of a compact space is compact, the [extreme value theorem](https://bliptext.com/articles/extreme-value-theorem) holds for such spaces: a [continuous](https://bliptext.com/articles/continuous-function-topology) real-valued function on a nonempty compact space is bounded above and attains its supremum.\n(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](https://bliptext.com/articles/proper-map) is compact.\n\n## Compactifications\nEvery topological space X is an open [dense subspace](https://bliptext.com/articles/dense-topological-subspace) of a compact space having at most one point more than X, by the [Alexandroff one-point compactification](https://bliptext.com/articles/alexandroff-one-point-compactification).\nBy the same construction, every [locally compact](https://bliptext.com/articles/locally-compact) Hausdorff space X is an open dense subspace of a compact Hausdorff space having at most one point more than X.\n\n## Ordered compact spaces\nA nonempty compact subset of the [real numbers](https://bliptext.com/articles/real-number) has a greatest element and a least element.\nLet X be a [simply ordered](https://bliptext.com/articles/total-order) set endowed with the [order topology](https://bliptext.com/articles/order-topology).\nThen X is compact if and only if X is a [complete lattice](https://bliptext.com/articles/complete-lattice) (i.e. all subsets have suprema and infima).\n\n# Examples\nS1\nS2\n. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.\nis compact.\nThis follows from the [Heine–Borel theorem](https://bliptext.com/articles/heine-borel-theorem).\nThe open interval is not compact: the [open cover](https://bliptext.com/articles/open-cover) for\nn = 3, 4, ...\ndoes not have a finite subcover.\nSimilarly, the set of [rational numbers](https://bliptext.com/articles/rational-numbers) in the closed interval is not compact: the sets of [rational numbers](https://bliptext.com/articles/rational-numbers) in the intervals cover all the rationals in [0, 1] for\nn = 4, 5, ...\nbut this cover does not have a finite subcover.\nHere, the sets are open in the subspace topology even though they are not open as subsets of \\mathbb{R}.\n, where n takes all integer values in\nZ\n, cover \\mathbb{R} but there is no finite subcover.\nf : \\mathbb{R} → [0, 1]\nfrom the real number line to the closed unit interval, and define a topology on K so that a sequence \\{f_n\\} in K converges towards\nf ∈ K\nif and only if \\{f_n(x)\\} converges towards\nf(x)\nfor all real numbers x. There is only one such topology; it is called the topology of [pointwise convergence](https://bliptext.com/articles/pointwise-convergence) or the [product topology](https://bliptext.com/articles/product-topology).\nThen K is a compact topological space; this follows from the [Tychonoff theorem](https://bliptext.com/articles/tychonoff-theorem).\nf : →\nsatisfying the [Lipschitz condition](https://bliptext.com/articles/lipschitz-condition)\n≤\nfor all\nx, y ∈\n. Consider on K the metric induced by the [uniform distance](https://bliptext.com/articles/uniform-convergence) Then by the Arzelà–Ascoli theorem the space K is compact.\n\n## Algebraic examples","revisionCount":0},"hasMore":false,"nextCursor":null}

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