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Encyclopedia > Compact space

In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. 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). Euclid, detail from The School of Athens by Raphael. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... 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. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...


A more modern approach is to call a topological space compact if each of its open covers has a finite subcover. The Heine–Borel theorem affirms that this coincides with "closed and bounded" for subsets of Euclidean space. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematical analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following are equivalent: S is closed and bounded every open cover of S has a finite subcover, that is, S is compact. ...


Note: Some authors such as Bourbaki use the term "quasi-compact" instead and reserve the name "compact" for topological spaces that are Hausdorff and compact. Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

Contents


History and motivation

The term compact was introduced by Fréchet in 1906. Maurice Fréchet (born September 2, 1878, died June 4, 1973) was a French mathematician. ... 1906 (MCMVI) was a common year starting on Monday (see link for calendar). ...


It has long been recognized that a property like compactness is necessary to prove a lot of useful theorems. It used to be that "compact" meant "sequentially compact" (every sequence has a convergent subsequence). This was when primarily metric spaces were studied. The "covering compact" definition surpassed it because it allows us to consider the general topological space, and many of the old results about metric spaces can be generalized. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...


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

  • Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. Then we can 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. 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. The proof can be "rescued", however, if A is compact: we simply take a finite subcover of the cover {V(a)} of A. 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. Many of the arguments and results involving compact spaces follow such a pattern. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... 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 a certain way. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... 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. ...


Definitions

Compactness of subsets of Rn

For any subset of Euclidean space Rn, the following four conditions are equivalent: A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

  • Every open cover has a finite subcover. This is the definition most commonly used.
  • Every sequence in the set has a convergent subsequence, the limit point of which belongs to the set.
  • Every infinite subset of the set has an accumulation point in the set.
  • The set is closed and bounded. This is the condition that is easiest to verify, for example a closed interval or closed n-ball.

In other spaces, these conditions may or may not be equivalent, depending on the properties of the space. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... For a discussion of convergence and convergent series, see limit (mathematics). ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... 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. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...


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. Thus, compactness is a topological property. In a sense, the closed unit interval [0,1] is intrinsically compact, regardless of how it is embedded in R or Rn. This is a glossary of some terms used in the branch of mathematics known as topology. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...


The general definition goes as follows. A topological space X is called compact iff all its open covers have a finite subcover. Formally, this means that ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

for every arbitrary collection {U_i}_{iin I} of open subsets of X such that cup_{iin I} U_i = X, there is a finite subset Jsubset I such that cup_{iin J} U_i = 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. This definition is dual to the usual one stated in terms of open sets. In topology, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty. ...


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 is a topological space in which points can be separated by neighbourhoods. ...


Examples of compact spaces

In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... The title given to this article is incorrect due to technical limitations. ... This word should not be confused with homomorphism. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ... In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In mathematics, the Hilbert cube is a topological space that provides an instructive example of some ideas in topology. ... In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In topology, Sierpiński space S is the simplest example of a topological space that does not satisfy the T1 axiom. ...

Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions): This is a glossary of some terms used in the branch of mathematics known as topology. ...

  • A continuous image of a compact space is compact.
  • The extreme value theorem: a real continuous function on a compact space is bounded and attains its maximum.
  • A closed subset of a compact space is compact.
  • A compact subset of a Hausdorff space is closed.
  • A nonempty compact subset of the real numbers has a greatest element and a least element.
  • 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.
  • The product of any collection of compact spaces is compact. (Tychonoff's theorem -- this is equivalent to the axiom of choice)
  • A compact Hausdorff space is normal.
  • Every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
  • A metric space (or more generally any first countable space) is compact if and only if every sequence in the space has a convergent subsequence.
  • A topological space is compact if and only if every net on the space has a convergent subnet.
  • A topological space is compact if and only if every filter on the space has a convergent refinement.
  • A topological space is compact if and only if every ultrafilter on the space is convergent.
  • A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.
  • Every topological space X is a dense subspace of a compact space which has at most one point more than X. (Alexandroff one-point compactification)
  • A metric space X is compact if and only if every metric space homeomorphic to X is complete.
  • 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. (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. (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 topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In calculus, the extreme value theorem states that if a function f(x) is continuous in the closed interval [a,b] then f(x) must attain its maximum and minimum value, each at least once. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematical analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following are equivalent: S is closed and bounded every open cover of S has a finite subcover, that is, S is compact. ... In mathematics, a metric space is a set where a notion of distance 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 a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In mathematics, Tychonoffs 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, and T5 spaces are particularly nice kinds of topological spaces. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In mathematics, a filter is a special subset of a partially ordered set. ... In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology). ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... This word should not be confused with homomorphism. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

Other forms of compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

  • Sequentially compact: Every sequence has a convergent subsequence.
  • 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.
  • Weakly countably compact (or limit point compact): Every infinite subset has an accumulation point.

While all these conditions are equivalent for metric spaces, in general we have the following implications: In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... Partial plot of a function f. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

  • Compact spaces are countably compact.
  • Sequentially compact spaces are countably compact.
  • Countably compact spaces are pseudocompact and weakly countably compact.

Not every countably compact space is compact; an example is given by the first uncountable ordinal with the order topology. Not every compact space is sequentially compact; an example is the infinite product space 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. For complete metric spaces this is equivalent to compactness. See relatively compact for the topological version. In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ...


Another related notion that is usually 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 like a small portion of a compact space. ...


See also

In mathematics, especially analysis, exhaustion by compact sets of an open set E in the Euclidean space Rn (or a manifold with countable base) is an increasing sequence of compact sets , where by increasing we mean is a subset of , with the limit (union) of the sequence being E. Sometimes...

References


  Results from FactBites:
 
Compact space - Wikipedia, the free encyclopedia (1380 words)
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).
A compact subset of a Hausdorff space is closed.
Countably compact spaces are pseudocompact and weakly countably compact.
Compact space - definition of Compact space in Encyclopedia (1376 words)
In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R
An equivalent definition of compact spaces, sometimes useful, is based on the finite intersection property.
A metric space is compact if and only if every sequence in the space has a convergent subsequence.
  More results at FactBites »

 
 

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