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Topic: Compact Hausdorff space

Related Topics

Hausdorff space

Separable metric space

Hilbert cube

Delta function

Complex number

	Hausdorff space - Wikipedia, the free encyclopedia
		Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology.
		Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces.
		Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.
	en.wikipedia.org /wiki/Hausdorff_space (1227 words)

	Tychonoff space - Wikipedia, the free encyclopedia
		X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F.
		More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K and an injective continuous map j from X to K such that the inverse of j is also continuous.
		It is characterised by the universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a unique continuous map g from βX to Y that extends f in the sense that f is the composition of g and j.
	www.wikipedia.com /wiki/Tychonov_space (600 words)

	Compact space (Site not responding. Last check: 2007-08-13)
		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.
		Countably compact spaces are pseudocompact and weakly countably compact.
	hallencyclopedia.com /Compact_space (1634 words)

	Locally compact space - Wikipedia, the free encyclopedia
		Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article Compact space.
		Thus locally compact spaces are as useful in p-adic analysis as in classical analysis.
		The notion of local compactness is important in the study of topological groups mainly because every locally compact Hausdorff group G carries natural measures called the Haar measures which allow one to integrate functions defined on G.
	en.wikipedia.org /wiki/Locally_compact (1326 words)

	Compact space - Wikipedia, the free encyclopedia
		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).
		Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x.
		A compact subset of a Hausdorff space is closed.
	en.wikipedia.org /wiki/Compact_space (1265 words)

	PlanetMath: examples of compact spaces
		is a compact space, a consequence of Tychonoff's theorem.
		Any profinite group is compact Hausdorff: finite discrete spaces are compact Hausdorff, therefore their product is compact Hausdorff, and a profinite group is a closed subset of such a product.
		This is version 13 of examples of compact spaces, born on 2002-06-25, modified 2003-09-12.
	planetmath.org /encyclopedia/ExamplesOfCompactSpaces.html (395 words)

	Compact space - Wikipedia
		A compact space is a topological space in which every open cover has a finite subcover.
		Some authors reserve the term 'compact' for compact Hausdorff spaces, but this article follows the usual current practice of allowing compact spaces to be non-Hausdorff.
		A metric space is compact if and only if it is complete and totally bounded.
	nostalgia.wikipedia.org /wiki/Compact_space (384 words)

	Topology - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-08-13)
		In 1914, Felix Hausdorf, generalizing the notion of metric space, coined the term "topological space" and gave the definition for what is now called Hausdorff space.
		The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
		A compact subspace of a Hausdorff space is closed.
	encyclopedia.learnthis.info /t/to/topology_1.html (1288 words)

	Free Encyclopedia (Site not responding. Last check: 2007-08-13)
		The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated.
		Further, there is a generalization of the Stone-Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space space is approximated uniformly on compact sets by algebras of the type appearing in the Stone-Weierstrass theorem and described below.
		Suppose K is a compact Hausdorff space with at least two points and L is a lattice in C(K,R).
	www.freeencyclopedia.net /index.php?title=Stone-Weierstrass_theorem (938 words)

	Closed set - Iridis Encyclopedia (Site not responding. Last check: 2007-08-13)
		In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets.
		However, the compact Hausdorff spaces are "absolutely closed" in a certain sense.
		Stone-Čech compactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.
	www.iridis.com /Closed (503 words)

	Locally compact space Information - TextSheet.com (Site not responding. Last check: 2007-08-13)
		To be precise, a topological space X is locally compact iff every point has a local base of compact neighborhoodss.
		Since every locally compact Hausdorff space X is Tychonoff, it can be embedded in a compact Hausdorff space b(X) using the Stone-Cech compactification.
		More precisely, the categories of locally compact Hausdorff spaces and of commutative C* algebras are dual; this is the Gelfand-Naimark theorem.
	www.rsk.soldat.sferahost.com /encyclopedia/l/lo/locally_compact_space.html (1295 words)

	PlanetMath: a compact set in a Hausdorff space is closed
		A compact set in a Hausdorff space is closed.
		"a compact set in a Hausdorff space is closed" is owned by mathcam.
		This is version 3 of a compact set in a Hausdorff space is closed, born on 2003-04-17, modified 2003-05-03.
	planetmath.org /encyclopedia/ACompactSetInAHausdorffSpaceIsClosed.html (189 words)

	Bohr compactification - Wikipedia, the free encyclopedia
		In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G.
		Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr(G) and a continuous homomorphism
		The kernel of b consists exactly of those elements of G which cannot be separated from the identity of G by finite-dimensional unitary representations.
	www.wikipedia.org /wiki/Bohr_compactification (324 words)

	An Introduction to Banach Space Theory
		Section 1.12, devoted to separability, includes the Banach-Mazur characterization of separable Banach spaces as isomorphs of quotient spaces of \ell_1, and ends with the characterization of separable normed spaces as the normed spaces that are compactly generated so that the stage is set for the introduction of weakly compactly generated normed spaces in Section 2.8.
		The goal of optional Section 2.9 is to obtain James's characterization of weakly compact subsets of a Banach space in terms of the behavior of bounded linear functionals.
		Schauder's theorem relating the compactness of a bounded linear operator to that of its adjoint is presented, as is the characterization of operator compactness in terms of the bounded-weak*-to-norm continuity of the adjoint.
	www.math.lsa.umich.edu /~meggin/ibst.html (2875 words)

	Locally Compact [Definition] (Site not responding. Last check: 2007-08-13)
		The Hausdorff condition is one in a series of separation axioms that can be imposed on a topological space, however it is the one that is most frequently used and discussed....
		Thus locally compact spaces are as useful in p-adic analysis P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers.
		The notion of local compactness is important in the study of topological groups mainly because every locally compact Hausdorff group G carries natural measuresIn mathematics, a measure is a function that assigns a number, e.g., a "size", "volume", or "probability", to subsets of a given set.
	www.wikimirror.com /Locally_compact (3529 words)

	PlanetMath: point and a compact set in a Hausdorff space have disjoint open neighborhoods.
		PlanetMath: point and a compact set in a Hausdorff space have disjoint open neighborhoods.
		"point and a compact set in a Hausdorff space have disjoint open neighborhoods." is owned by drini.
		This is version 8 of point and a compact set in a Hausdorff space have disjoint open neighborhoods.
	planetmath.org /encyclopedia/APointAndACompactSetInAHausdorffSpaceHaveDisjointOpenNeighborhoods.html (163 words)

	[No title] (Site not responding. Last check: 2007-08-13)
		This is an example of an "extremely disconnected space": the closure of every open set is open.
		Every totally disconnected compact Hausdorff space arises as an example of (8), up to homeomorphism, according to the Stone Representation Theorem for Boolean algebras.
		Given a totally disconnected compact Hausdorff space X, one can recover the boolean algebra as the boolean agebra of simultaneously open and closed subsets of X. This is explained in Halmos' nice book Lectures on Boolean Algebras, which is where I learned most of what I know about it.
	www.math.niu.edu /~rusin/known-math/00_incoming/disconn (316 words)

	PlanetMath: proof of A compact set in a Hausdorff space is closed
		is compact, this open cover admits a finite subcover.
		"proof of A compact set in a Hausdorff space is closed" is owned by jay.
		This is version 1 of proof of A compact set in a Hausdorff space is closed, born on 2003-04-23.
	planetmath.org /encyclopedia/ProofOfACompactSetInAHausdorffSpaceIsClosed.html (159 words)

	Stone-Cech Compactification - NoiseFactory Science Archives (http://noisefactory.co.uk) (Site not responding. Last check: 2007-08-13)
		Compact spaces are very popular with topologists, because they're very easy to reason about.
		If X is one of the standard spaces used in complicated mathematics it probably isn't compact, and that means it may be quite hard to reason about its properties.
		Consequently a space has a compactification (and so is Tychonov) if and only if it can be identified with a subspace of a compact Hausdorff space.
	noisefactory.co.uk /maths/stone-cech.html (1649 words)

	Current Seminars
		Spaces with the property of the title are called almost SV-spaces.
		The first such spaces to be studied extensively were called F-spaces and are characterized by having all cozerosets C*-embedded.
		Those who hope that further development of analysis on Scott domains would, for example, enable the use of standard optimization techniques on the spaces of programs are, so far, disappointed.
	home.att.net /~topann/CurrSem.html (508 words)

	Metrization theorems Article, Metrizationtheorems Information (Site not responding. Last check: 2007-08-13)
		A metrizable space is a topological space that is homeomorphic to a metric space.
		This states that everysecond-countable regular Hausdorff space is metrizable.
		For example, a compact Hausdorff space is metrizable if and only if it is second-countable.
	www.anoca.org /metrizable/space/metrization_theorems.html (313 words)

	Noncommutative geometry (Site not responding. Last check: 2007-08-13)
		In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them.
		This is by analogy with the Gelfand representation, which shows that commutative C-algebras are dual to locally compact* Hausdorff spaces.
		The symplectic phase space of classical mechanics is deformed into a non-commutative phase space generated by the position and momentum operators.
	www.worldhistory.com /wiki/N/Noncommutative-geometry.htm (585 words)

	Untitled Document
		The corresponding knots in the lens spaces are $0$- or $1$-bridge braids with respect to the Heegaard torus of the lens space.
		Abstract: C^-algberas are a special class of norm-closed algerbas of operators on Hilbert space* that have special connections with in topology, geometry, harmonic analysis, etc. Any COMMUTATIVE C-algebra is naturally isomorphic to the algebra of complex functions on a compact* Hausdorff space naturally associated with it.
		We extend this (commutative) Gelfand-Naimark duality to the general (possibly noncommutative) C^-algebras, by identifying the noncommutative analog of compact* Hausdorff spaces.
	www.math.uga.edu /seminars_conferences/11.29.04.html (502 words)

	[No title] (Site not responding. Last check: 2007-08-13)
		BX is compact Hausdorff, of course, and B is a functor from spaces to compact Hausdorff spaces.
		X is a subspace of a normal space d.
		X is a subspace of a compact Hausdorff space Note that regularity and complete regularity are (obviously) inherited by subspaces, but normality is not.
	www.lehigh.edu /dmd1/public/www-data/tg1013.txt (486 words)

	Mathematics Colloquium (Site not responding. Last check: 2007-08-13)
		In 1890's Poincare' brought the geometry of the phase space (space of "possible values") to bear on the analysis to investigate the integral curves in their entire domain of existence (Poincare' reccurence theorem).
		Thus, the focus shiffted away from differential equations that define a dynamical system to the phase space and the group of transformations implicit in the system.
		We shall focus on topological dynamical systems where a discrete semigroup is acting on a compact Hausdorff space.
	math.dartmouth.edu /~colloq/f96/franek.html (350 words)

	Atlas: Spaces in which Point-finite Open Covers have Finite Subcovers by Nurettin Ergun
		Another related well known result has been proved by Iseki and Kasahara in 1957: A regular Hausdorff space X is countably compact if and only if every point-finite open cover of X has a finite subcover.
		He defined a non countably compact Hausdorff space in which point-finite open covers have finite subcovers.
		In fact any countably compact Hausdorff space is collectionwise Hausdorff since closed-discrete subsets are necessarily finite in such spaces, but, examples of non-regular countably compact Hausdorff spaces are already known.
	atlas-conferences.com /cgi-bin/abstract/caeh-05 (521 words)

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