Compact space

"Compact" redirects here. For other uses, see Compact (disambiguation).

In mathematics, a subset of Euclidean space Rⁿ is called compact if it is closed and bounded. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematical analysis and related areas of Mathematics, a set is called bounded, if it is in a certain sense of finite size For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed). In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

A more modern approach is to call a topological space compact if each of its open covers has a finite subcover. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. The Heine–Borel theorem shows that this definition is equivalent to "closed and bounded" for subsets of Euclidean space. In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset Note: Some authors such as Bourbaki use the term "quasi-compact" instead, and reserve the term "compact" for topological spaces that are Hausdorff and "quasi-compact". Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

A single compact set is sometimes referred to as a compactum; following the Latin second declension (neuter), the corresponding plural form is compacta. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Latin is an inflected language and as such its nouns pronouns and adjectives must be declined in order to serve a grammatical function

1 History and motivation
2 Definitions
- 2.1 Compactness of subsets of Rⁿ
- 2.2 Compactness of topological spaces
3 Examples of compact spaces
4 Theorems
5 Other forms of compactness
6 See also
7 Notes
8 References

History and motivation

The term compact was introduced by Fréchet in 1906. Maurice Fréchet ( September 2, 1878 – June 4, 1973) was a French Mathematician. Year 1906 ( MCMVI) was a Common year starting on Monday (link will display full calendar of the Gregorian calendar (or a Common year starting

It has long been recognized that a property like compactness is necessary to prove many useful theorems. It used to be that "compact" meant "sequentially compact" (every sequence has a convergent subsequence). In Mathematics, a sequence is an ordered list of objects (or events This was when primarily metric spaces were studied. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined The "covering compact" definition has become more prominent because it allows us to consider general topological spaces, and many of the old results about metric spaces can be generalized to this setting. This generalization is particularly useful in the study of function spaces, many of which are not metric spaces. In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y.

One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. It is often said that "compactness is the next best thing to finiteness". Here is an example:

Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space Then we can separate x and A by neighbourhoods: for each a in A, let U(x) and V(a) be disjoint neighbourhoods containing x and a, respectively. In Topology and related branches of Mathematics, separated sets are pairs of Subsets of a given Topological space that are related to each other In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A.

Note that if A is infinite, the proof fails, because the intersection of arbitrarily many neighbourhoods of x might not be a neighbourhood of x. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The proof can be "rescued", however, if A is compact: we simply take a finite subcover of the cover {V(a) : a in A} of A, then intersect the corresponding finitely many U(x). In this way, we see that in a Hausdorff space, any point can be separated by neighbourhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighbourhoods -- note that this is precisely what we get if we replace "point" (i. e. singleton set) with "compact set" in the Hausdorff separation axiom. In Mathematics, a singleton is a set with exactly one element In Topology and related fields of Mathematics, there are several restrictions that one often makes on the kinds of Topological spaces that one wishes to consider Many of the arguments and results involving compact spaces follow such a pattern.

Definitions

Compactness of subsets of Rⁿ

For any subset of Euclidean space Rⁿ, the following four conditions are equivalent:

Every open cover has a finite subcover. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection This is the most commonly used definition.
Every sequence in the set has a convergent subsequence, the limit point of which belongs to the set. In Mathematics, a sequence is an ordered list of objects (or events In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or
Every infinite subset of the set has an accumulation point in the set. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"
The set is closed and bounded. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematical analysis and related areas of Mathematics, a set is called bounded, if it is in a certain sense of finite size This is the condition that is easiest to verify, for example a closed interval or closed n-ball. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

In other spaces, these conditions may or may not be equivalent, depending on the properties of the space.

Note that while compactness is a property of the set itself (with its topology), closedness is relative to a space it is in; above "closed" is used in the sense of closed in Rⁿ. A set which is closed in e. g. Qⁿ is typically not closed in Rⁿ, hence not compact.

Compactness of topological spaces

The "finite subcover" property from the previous paragraph is more abstract than the "closed and bounded" one, but it has the distinct advantage that it can be given using the subspace topology on a subset of Rⁿ, eliminating the need of using a metric or an ambient space. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Thus, compactness is a topological property. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is In a sense, the closed unit interval [0,1] is intrinsically compact, regardless of how it is embedded in R or Rⁿ.

A topological space X is defined as compact if all its open covers have a finite subcover. Formally, this means that

for every arbitrary collection $\{U_\alpha\}_{\alpha\in A}$ of open subsets of

X

such that $\bigcup_{\alpha\in A} U_\alpha \supseteq X$ , there is a finite subset $J\subset A$ such that $\bigcup_{i\in J} U_i \supseteq X$ .

An often used equivalent definition is given in terms of the finite intersection property: if any collection of closed sets satisfying the finite intersection property has nonempty intersection, then the space is compact^[1]. In General topology, the finite intersection property is a property of a collection of subsets of a set X. This definition is dual to the usual one stated in terms of open sets.

Some authors require that a compact space also be Hausdorff, and the non-Hausdorff version is then called quasicompact. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

Examples of compact spaces

Any finite topological space, including the empty set, is compact. In Mathematics, a finite topological space is a Topological space for which the underlying point set is finite. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members Slightly more generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology. It is possible for a Topology to be finite in the sense that there are onlyfinitely many open sets
The closed unit interval $[0,1]$ is compact. In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x This follows from the Heine-Borel theorem; the proof of which is about as hard as proving directly that $[0,1]$ is compact. In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset The open interval $(0,1)$ is not compact: the open cover $(1 / n,1 - 1 / n)$ for $n = 3,4,. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection . .$ does not have a finite subcover.
For every natural number n, the n-sphere is compact. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Again from the Heine-Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to 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. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed
The Cantor set is compact. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith Since the p-adic integers are homeomorphic to the Cantor set, they also form a compact set. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 Topological equivalence redirects here see also Topological equivalence (dynamical systems. Since a finite set containing p elements is compact, this shows that the countable product of finite sets is compact, and is thus a special case of Tychonoff's theorem. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Mathematics, Tychonoff's theorem states that the product of any collection of compact Topological spaces is compact
Consider the set $K$ of all functions $f: \mathbb{R} \rightarrow [0,1]$ from 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 $f\in K$ if and only if ${f n (x)}$ converges towards $f (x)$ for all $x\in\mathbb{R}$ . There is only one such topology; it is called the topology of pointwise convergence. In Mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function Then $K$ is a compact topological space, again a consequence of Tychonoff's theorem.
Consider the set $K$ of all functions $f\colon [0,1]\to [0,1]$ satisfying the Lipschitz condition $|f(x)-f(y)|\le |x-y|$ for all $x,y\in[0,1]$ and consider on $K$ the metric induced by the uniform distance $d(f,g)=\sup\{|f(x)-g(x)| \colon x\in [0,1]\}$ . In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions In the mathematical field of analysis, uniform convergence is a type of Convergence stronger than Pointwise convergence. Then by Ascoli-Arzelà theorem the space $K$ is compact. In Mathematics, the Arzelà–Ascoli theorem of Functional analysis gives necessary and sufficient conditions to decide whether a set of Continuous functions
Any space carrying the cofinite topology is compact. In Mathematics, a cofinite Subset of a set X is a subset Y whose complement in X is a finite set
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. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks The one-point compactification of $\mathbb{R}$ is homeomorphic to the circle $S 1$ ; the one-point compactification of $\mathbb{R}^2$ is homeomorphic to the sphere $S 2$ . Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
The spectrum of any continuous linear operator on a Hilbert space is a compact subset of the complex numbers C. In Functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of Eigenvalues for matrices In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This article assumes some familiarity with Analytic geometry and the concept of a limit. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted If the Hilbert space is infinite-dimensional, then any compact subset of C arises in this manner, as the spectrum of some continuous linear operator on the Hilbert space.
The spectrum of any commutative ring or Boolean algebra is compact. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
The Hilbert cube is compact. In Mathematics, the Hilbert cube, named after David Hilbert, is a Topological space that provides an instructive example of some ideas in Topology
The right order topology or left order topology on any bounded totally ordered set is compact. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In particular, Sierpinski space is compact. In Mathematics, Sierpiński space (or the connected two-point set) is a Finite topological space with two points only one of which is closed
The prime spectrum of any commutative ring with the Zariski topology is a compact space, important in algebraic geometry. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with These prime spectra are almost never Hausdorff spaces. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions):

A continuous image of a compact space is compact. This is a glossary of some terms used in the branch of Mathematics known as Topology. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function ^[2]
The extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded and attains its supremum. In Calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed interval, then f
A closed subset of a compact space is compact. ^[3]
A compact subset of a Hausdorff space is closed. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space
A nonempty compact subset of the real numbers has a greatest element and a least element. In Mathematics, the real numbers may be described informally in several different ways
A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine–Borel theorem)
A metric space (or uniform space) is compact if and only if it is complete and totally bounded. In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any
The product of any collection of compact spaces is compact. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural (Tychonoff's theorem, which is equivalent to the axiom of choice)
A compact Hausdorff space is normal. In Mathematics, Tychonoff's theorem states that the product of any collection of compact Topological spaces is compact In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces
Every continuous map from a compact space to a Hausdorff space is closed and proper. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Mathematics, a Continuous function between Topological spaces is called proper if Inverse images of compact subsets are compact It follows that every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Topological equivalence redirects here see also Topological equivalence (dynamical systems.
A metric space (or more generally any first-countable uniform space) is compact if and only if every sequence in the space has a convergent subsequence. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In Mathematics, a sequence is an ordered list of objects (or events
A topological space is compact if and only if every net on the space has a convergent subnet. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics
A topological space is compact if and only if every filter on the space has a convergent refinement. In Mathematics, a filter is a special Subset of a Partially ordered set.
A topological space is compact if and only if every ultrafilter on the space is convergent. In the mathematical field of Set theory, an ultrafilter on a set X is a collection of Subsets of X that is a filter, that
A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space. In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces
Every non-compact topological space X is a dense subspace of a compact space which has at most one point more than X. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is (Alexandroff one-point compactification)
If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact. (Lebesgue's number lemma)
If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact. In Topology, Lebesgue's number lemma states If the Metric space (X d is compact and an Open cover of X (Alexander's sub-base theorem)
Two compact Hausdorff spaces X₁ and X₂ are homeomorphic if and only if their rings of continuous real-valued functions C(X₁) and C(X₂) are isomorphic. In Highway engineering, subbase is a layer between Subgrade and the Base course. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication (Gelfand-Naimark theorem)

Other forms of compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. In Mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of Bounded operators In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined These include the following.

Sequentially compact: Every sequence has a convergent subsequence. In Mathematics, a sequence is an ordered list of objects (or events
Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point. )
Pseudocompact : Every real-valued continuous function on the space is bounded. In Mathematics, in the field of Topology, a Topological space is said to be pseudocompact if its image under any Continuous function to In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function
Weakly countably compact (or limit point compact): Every infinite subset has an accumulation point. In Mathematics, particularly Topology, limit point compactness is a certain condition on a Topological space which generalizes some features of compactness In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"

While all these conditions are equivalent for metric spaces, in general we have the following implications:

Compact spaces are countably compact. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined
Sequentially compact spaces are countably compact.
Countably compact spaces are pseudocompact and weakly countably compact.

Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology. Not every compact space is sequentially compact; an example is given by 2^[0,1], with the product topology.

A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces. In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any In the Mathematical field of Topology, a uniform space is a set with a uniform structure. For complete metric spaces this is equivalent to compactness. See relatively compact for the topological version. In Mathematics, a relatively compact subspace (or relatively compact subset) Y of a Topological space X is a subset whose closure

Another related notion which (by most definitions) is strictly weaker than compactness is local compactness. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks

Notes

References

Lynn Arthur Steen and J. In Mathematics, especially analysis, exhaustion by compact sets of an Open set E in the Euclidean space R n In Mathematics, in the field of General topology, a Topological space is said to be metacompact if every Open cover has a Point finite In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. Lynn Arthur Steen is an American Mathematician who is Professor of Mathematics at St Arthur Seebach, Jr. , Counterexamples in Topology (1978) Springer-Verlag, New York
Countably compact on PlanetMath

This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the GFDL. Counterexamples in Topology (1970 2nd ed 1978 is a book on Mathematics by topologists Lynn Steen and J PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(mathematics) Any topological subset of Euclidean space that is a compact set

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