Compact space

compactcompact setcompactnesscompact subsetcompact Hausdorff spacenoncompactquasi-compactcompact subsetscompact topological spacecompact sets
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Unit interval

closed unit interval[0,1interval
Thus, if one chooses an infinite number of points in the closed unit interval A non-trivial example of a compact space is the (closed) unit interval of real numbers.
As a topological space, it is compact, contractible, path connected and locally path connected.

Extreme value theorem

boundedboundedness theoremboundedness, ultimate boundedness
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.
In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.

Bounded set

boundedunboundedbounded subset
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
A metric space is compact if and only if it is complete and totally bounded.

Limit point compact

weakly countably compact
Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces.
This property generalizes a property of compact spaces.

Open set

openopen subsetopen sets
The most useful notion, which is 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.
Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets.

Maurice René Fréchet

FréchetMaurice FréchetFréchet, Maurice René
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 which led to the famous 1906 thesis).
His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness.

Arzelà–Ascoli theorem

Arzelà-Ascoli theorem
A typical application is furnished by the Arzelà–Ascoli theorem or 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. 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.
A further generalization of the theorem was proven by, to sets of real-valued continuous functions with domain a compact metric space.

General topology

point-set topologypoint set topologytopology
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
Compact sets are those that can be covered by finitely many sets of arbitrarily small size.

Sequentially compact space

sequentially compactsequential compactnesssequentially'' compact
Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric 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.
Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (in the topology induced by the metric).

Heine–Borel theorem

Heine-Borel theoremHeine–Borel propertyHeine-Borel property
The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
S is compact, that is, every open cover of S has a finite subcover.

Metric space

metricmetric spacesmetric geometry
Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces.
This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers.

Pavel Urysohn

UrysohnP. UrysohnPaul Urysohn
This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets.
He and Pavel Alexandrov formulated the modern definition of compactness in 1923.

Hilbert space

Hilbertseparable Hilbert spacecomplex Hilbert space
For a certain class of 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.
Using these methods on a compact Riemannian manifold, one can obtain for instance the Hodge decomposition, which is the basis of Hodge theory.

Closed set

closedclosed subsetclosed sets
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
However, the compact Hausdorff spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here.

Finite intersection property

strong finite intersection propertycompactness
4) Any collection of closed subsets of X with the finite intersection property has nonempty intersection.
The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets having the finite intersection property has nonempty intersection.

Topological space

topologytopological spacestopological structure
This notion is defined for more general topological spaces than Euclidean space in various ways.
Examples of such properties include connectedness, compactness, and various separation axioms.

Disk (mathematics)

diskdiscdisks
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.
For instance, every closed disk is compact whereas every open disk is not compact.

Axiom of choice

ChoiceACaxiom of non-choice
Assuming the axiom of choice, the following are equivalent:
Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

Finite topological space

finite subspacesfinite spacefinite spaces
Any finite space is trivially compact.
Every finite topological space is compact since any open cover must already be finite.

Lindelöf space

LindelöfLindelöf numberLindelöf property
2) (X, d) 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.
The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.

Uniform convergence

uniformlyuniformly convergentconverges uniformly
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.
If S is a compact interval (or in general a compact topological space), and (f_n) is a monotone increasing sequence (meaning for all n and x) of continuous functions with a pointwise limit f which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if S is a compact interval and (f_n) is an equicontinuous sequence that converges pointwise.

Compactification (mathematics)

compactificationcompactifiedcompactifications
Every topological space X is an open dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.

Complete metric space

completecompletioncompleteness
2) (X, d) is complete and totally bounded (this is also equivalent to compactness for uniform spaces).
Every compact metric space is complete, though complete spaces need not be compact.

Real number

realrealsreal-valued
A non-trivial example of a compact space is the (closed) unit interval of real numbers.
The real numbers are locally compact but not compact.

Separable space

separableseparabilityseparability axiom
2) (X, d) 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.
The property of separability does not in and of itself give any limitations on the cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected.