Compact space

File:Compact.svg

Per the compactness criteria for Euclidean space as stated in the Heine–Borel theorem, the interval Template:Math is not compact because it is not bounded. The interval Template:Math is not compact because it is not closed (but bounded). The interval Template:Math is compact because it is both closed and bounded.

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.<ref>Template:Cite encyclopedia</ref> The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers <math>\mathbb{Q}</math> is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers <math>\mathbb{R}</math> is not compact either, because it excludes the two limiting values <math>+\infty</math> and <math>-\infty</math>. However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.<ref>Template:Cite book</ref> The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval Template:Math, some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence Template:Nobr accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval Template:Math, those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering <math>\mathbb{R}^1</math> (the real number line), the sequence of points Template:Nobr has no subsequence that converges to any real number.

Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In general topological spaces, however, these notions of compactness are not necessarily equivalent. The 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 that "cover" the space, in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally – that is, in a neighborhood of each point – into corresponding statements that hold throughout the space, and many theorems are of this character.

The term compact set is sometimes used as a synonym for compact space, but also often refers to a compact subspace of a topological space.

Historical development

In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called 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>Template:Harvnb; Template:Harvnb</ref>

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for 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>Template:Harvnb</ref> The culmination of their investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a 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 equations, as investigated by David Hilbert and Erhard Schmidt. For a certain class of 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 – 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 Fréchet who, in 1906, 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<ref>Frechet, M. 1904. {{#invoke:Lang|lang}}. {{#invoke:Lang|lang}}.</ref> which led to the famous 1906 thesis).

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the 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 (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (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 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 Template:Harvtxt, who also exploited it in the development of the 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. Template:Harvtxt showed that the earlier version of compactness due to Fréchet, now called (relative) sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.

Basic examples

Any finite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) unit interval Template:Closed-closed of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point among these points in that interval. For instance, the odd-numbered terms of the sequence Template:Nowrap get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval of the real numbers is not compact. It is also crucial that the interval be bounded, since in the interval Template:Closed-open, one could choose the sequence of points Template:Nowrap, of which no sub-sequence ultimately gets arbitrarily close to any given real number.

In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary – without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

Definitions

Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.<ref name=":0"/>

In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness – originally called bicompactness – is defined using covers consisting of open sets (see Open cover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally – in a neighbourhood of each point of the space – and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.

Open cover definition

Formally, a topological space Template:Mvar is called compact if every open cover of Template:Mvar has a finite subcover.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> That is, Template:Mvar is compact if for every collection Template:Mvar of open subsets<ref>Here, "collection" means "set" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset".</ref> of Template:Mvar such that

<math display="block">X = \bigcup_{S \in C}S\ ,</math>

there is a finite subcollection Template:Mvar ⊆ Template:Mvar such that

<math display="block">X = \bigcup_{S \in F} S\ .</math>

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

Compactness of subsets

A subset Template:Mvar of a topological space Template:Mvar is said to be compact if it is compact as a subspace (in the subspace topology). That is, Template:Mvar is compact if for every arbitrary collection Template:Mvar of open subsets of Template:Mvar such that

<math display="block">K \subseteq \bigcup_{S \in C} S\ ,</math>

there is a finite subcollection Template:Mvar ⊆ Template:Mvar such that

<math display="block">K \subseteq \bigcup_{S \in F} S\ .</math>

Because compactness is a topological property, the compactness of a subset depends only on the subspace topology induced on it. It follows that, if <math>K \subset Z \subset Y</math>, with subset Template:Mvar equipped with the subspace topology, then Template:Mvar is compact in Template:Mvar if and only if Template:Mvar is compact in Template:Mvar.

Characterization

If Template:Mvar is a topological space then the following are equivalent:

1. Template:Mvar is compact; i.e., every open cover of Template:Mvar has a finite subcover.
2. Template:Mvar 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).
3. Template:Mvar is Lindelöf and countably compact.Template:Sfn
4. Any collection of closed subsets of Template:Mvar with the finite intersection property has nonempty intersection.
5. Every net on Template:Mvar has a convergent subnet (see the article on nets for a proof).
6. Every filter on Template:Mvar has a convergent refinement.
7. Every net on Template:Mvar has a cluster point.
8. Every filter on Template:Mvar has a cluster point.
9. Every ultrafilter on Template:Mvar converges to at least one point.
10. Every infinite subset of Template:Mvar has a complete accumulation point.<ref>Template:Harvnb</ref>
11. For every topological space Template:Mvar, the projection <math>X \times Y \to Y</math> is a closed mapping<ref name="Bourbaki">Template:Harvnb</ref> (see proper map).
12. Every open cover linearly ordered by subset inclusion contains Template:Mvar.Template:Sfn

Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).<ref name="BourbakiDefinition">Template:Harvnb</ref>

Euclidean space

For any subset Template:Mvar of Euclidean space, Template:Mvar is compact if and only if it is closed and bounded; this is the Heine–Borel theorem.

As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed Template:Mvar-ball.

Metric spaces

For any metric space Template:Math, the following are equivalent (assuming countable choice):

1. Template:Math is compact.
2. Template:Math is complete and totally bounded (this is also equivalent to compactness for uniform spaces).<ref>Template:Harvnb</ref>
3. Template:Math is sequentially compact; that is, every sequence in Template:Mvar has a convergent subsequence whose limit is in Template:Mvar (this is also equivalent to compactness for first-countable uniform spaces).
4. Template:Math is limit point compact (also called weakly countably compact); that is, every infinite subset of Template:Mvar has at least one limit point in Template:Mvar.
5. Template:Math is countably compact; that is, every countable open cover of Template:Mvar has a finite subcover.
6. Template:Math is an image of a continuous function from the Cantor set.<ref>Template:Harvnb Theorem 30.7.</ref>
7. Every decreasing nested sequence of nonempty closed subsets Template:Math in Template:Math has a nonempty intersection.
8. Every increasing nested sequence of proper open subsets Template:Math in Template:Math fails to cover Template:Mvar.

A compact metric space Template:Math also satisfies the following properties:

1. Lebesgue's number lemma: For every open cover of Template:Mvar, there exists a number Template:Nowrap such that every subset of Template:Mvar of diameter < Template:Mvar is contained in some member of the cover.
2. Template:Math is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.
3. Template:Mvar is closed and bounded (as a subset of any metric space whose restricted metric is Template:Mvar). The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.

Ordered spaces

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

1. Template:Math is compact.
2. Every subset of Template:Mvar has a supremum (i.e. a least upper bound) in Template:Mvar.
3. Every subset of Template:Mvar has an infimum (i.e. a greatest lower bound) in Template:Mvar.
4. Every nonempty closed subset of Template:Mvar has a maximum and a minimum element.

An ordered space satisfying (any one of) these conditions is called a complete lattice.

In addition, the following are equivalent for all ordered spaces Template:Math, and (assuming countable choice) are true whenever Template:Math is compact. (The converse in general fails if Template:Math is not also metrizable.):

1. Every sequence in Template:Math has a subsequence that converges in Template:Math.
2. Every monotone increasing sequence in Template:Mvar converges to a unique limit in Template:Mvar.
3. Every monotone decreasing sequence in Template:Mvar converges to a unique limit in Template:Mvar.
4. Every decreasing nested sequence of nonempty closed subsets Template:Mvar₁ ⊇ Template:Mvar₂ ⊇ ... in Template:Math has a nonempty intersection.
5. Every increasing nested sequence of proper open subsets Template:Mvar₁ ⊆ Template:Mvar₂ ⊆ ... in Template:Math fails to cover Template:Mvar.

Characterization by continuous functions

Let Template:Mvar be a topological space and Template:Math the ring of real continuous functions on Template:Mvar. For each Template:Math, the evaluation map <math>\operatorname{ev}_p\colon C(X)\to \mathbb{R}</math> given by Template:Math is a ring homomorphism. The kernel of Template:Math is a maximal ideal, since the residue field Template:Nowrap is the field of real numbers, by the first isomorphism theorem. A topological space Template:Mvar is pseudocompact if and only if every maximal ideal in Template:Math has residue field the real numbers. For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.<ref>Template:Harvnb</ref> There are pseudocompact spaces that are not compact, though.

In general, for non-pseudocompact spaces there are always maximal ideals Template:Mvar in Template:Math such that the residue field Template:Math is a (non-Archimedean) hyperreal field. The framework of non-standard analysis allows for the following alternative characterization of compactness:<ref>Template:Harvnb</ref> a topological space Template:Mvar is compact if and only if every point Template:Mvar of the natural extension Template:Math is infinitely close to a point Template:Math of Template:Mvar (more precisely, Template:Mvar is contained in the monad of Template:Math).

Hyperreal definition

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

Sufficient conditions

• A closed subset of a compact space is compact.<ref>Template:Harvnb</ref>
• A finite union of compact sets is compact.
• A continuous image of a compact space is compact.<ref>Template:Harvnb</ref>
• The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed);
  • If Template:Mvar is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).Template:Efn
• The product of any collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.)
• In a metrizable space, a subset is compact if and only if it is sequentially compact (assuming countable choice)
• A finite set endowed with any topology is compact.

Properties of compact spaces

• A compact subset of a Hausdorff space Template:Mvar is closed.
  • If Template:Mvar is not Hausdorff then a compact subset of Template:Mvar may fail to be a closed subset of Template:Mvar (see footnote for example).Template:Efn
  • If Template:Mvar is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).Template:Efn
• In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are not closed.
• If Template:Mvar and Template:Mvar are disjoint compact subsets of a Hausdorff space Template:Mvar, then there exist disjoint open sets Template:Mvar and Template:Mvar in Template:Mvar such that Template:Math and Template:Math.
• A continuous bijection from a compact space into a Hausdorff space is a homeomorphism.
• A compact Hausdorff space is normal and regular.
• If a space Template:Mvar is compact and Hausdorff, then no finer topology on Template:Mvar is compact and no coarser topology on Template:Mvar is Hausdorff.
• If a subset of a metric space Template:Math is compact then it is Template:Mvar-bounded.

Functions and compact spaces

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

Compactifications

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

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

Ordered compact spaces

A nonempty compact subset of the real numbers has a greatest element and a least element.

Let Template:Mvar be a simply ordered set endowed with the order topology. Then Template:Mvar is compact if and only if Template:Mvar is a complete lattice (i.e. all subsets have suprema and infima).<ref>Template:Harvnb</ref>

Examples

• Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
• Any space carrying the cofinite topology is compact.
• Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point compactification of <math>\mathbb{R}</math> is homeomorphic to the circle Template:Math; the one-point compactification of <math>\mathbb{R}^2</math> is homeomorphic to the sphere Template:Math. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
• The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpiński space is compact.
• No discrete space with an infinite number of points is compact. The collection of all singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
• In <math>\mathbb{R}</math> carrying the lower limit topology, no uncountable set is compact.
• In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not locally compact but is still Lindelöf.
• The closed unit interval Template:Math is compact. This follows from the Heine–Borel theorem. The open interval Template:Open-open is not compact: the open cover <math display="inline">\left( \frac{1}{n}, 1 - \frac{1}{n} \right)</math> for Template:Math does not have a finite subcover. Similarly, the set of rational numbers in the closed interval Template:Closed-closed is not compact: the sets of rational numbers in the intervals <math display="inline">\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 Template:Math but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of <math>\mathbb{R}</math>.
• The set <math>\mathbb{R}</math> 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 Template:Math, where Template:Mvar takes all integer values in Template:Math, cover <math>\mathbb{R}</math> but there is no finite subcover.
• On the other hand, the extended real number line carrying the analogous topology is compact; note that the cover described above would never reach the points at infinity and thus would not cover the extended real line. In fact, the set has the homeomorphism to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
• For every natural number Template:Mvar, the [[n-sphere|Template:Mvar-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.
• Consider the set Template:Mvar of all functions Template:Math from the real number line to the closed unit interval, and define a topology on Template:Mvar so that a sequence <math>\{f_n\}</math> in Template:Mvar converges towards Template:Math if and only if <math>\{f_n(x)\}</math> converges towards Template:Math for all real numbers Template:Mvar. There is only one such topology; it is called the topology of pointwise convergence or the product topology. Then Template:Mvar is a compact topological space; this follows from the Tychonoff theorem.
• A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded (Arzelà–Ascoli theorem).
• Consider the set Template:Mvar of all functions Template:Math satisfying the Lipschitz condition Template:Math for all Template:Math. Consider on Template:Mvar the metric induced by the uniform distance <math>d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|.</math> Then by the Arzelà–Ascoli theorem the space Template:Mvar is compact.
• The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers <math>\mathbb{C}</math>. Conversely, any compact subset of <math>\mathbb{C}</math> arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space <math>\ell^2</math> may have any compact nonempty subset of <math>\mathbb{C}</math> as spectrum.
• The space of Borel probability measures on a compact Hausdorff space is compact for the vague topology, by the Alaoglu theorem.
• A collection of probability measures on the Borel sets of Euclidean space is called tight if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures. Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.

Algebraic examples

• Topological groups such as an orthogonal group are compact, while groups such as a general linear group are not.
• Since the [[p-adic numbers|Template:Mvar-adic integers]] are homeomorphic to the Cantor set, they form a compact set.
• Any global field K is a discrete additive subgroup of its adele ring, and the quotient space is compact. This was used in John Tate's thesis to allow harmonic analysis to be used in number theory.
• The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact, but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact schemes, "quasi" referring to the non-Hausdorff nature of the topology.
• The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of profinite groups.
• The structure space of a commutative unital Banach algebra is a compact Hausdorff space.
• The Hilbert cube is compact, again a consequence of Tychonoff's theorem.
• A profinite group (e.g. Galois group) is compact.

See also

• Compactly generated space
• Compactness theorem
• Continuous functions on a compact Hausdorff space
• Eberlein compactum
• Exhaustion by compact sets
• Lindelöf space
• Metacompact space
• Noetherian topological space
• Orthocompact space
• Paracompact space
• Quasi-compact morphism
• Precompact set - also called totally bounded
• Relatively compact subspace
• Totally bounded

Bibliography

• Template:Cite journal
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• Template:Springer.
• Template:Cite book (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).
• Template:Cite journal
• Template:Cite book
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• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Howes Modern Analysis and Topology 1995
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite journal.
• Template:Cite book
• Template:Cite book

External links

• Template:Cite arXiv

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