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	Completely Hausdorff space - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-09)
		In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space.
		It follows that every completely Hausdorff space is Urysohn and every Urysohn space is Hausdorff.
		There are obscure examples of spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff.
	en.wikipedia.org /wiki/Completely_Hausdorff_space (442 words)

	Metrization theorem - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.
		Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is second-countable, regular and Hausdorff.
		uniformizability, a topological space homeomorphic to a uniform space
	en.wikipedia.org /wiki/Metrization_theorems (384 words)

	Normal space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		spaces are particularly nice kinds of ((mathematics) any set of points that satisfy a set of postulates of some kind) topological spaces.
		spaces are discussed elsewhere; they are related to (additional info and facts about paracompactness) paracompactness.
		Specifically, (additional info and facts about Sierpinski space) Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
	www.absoluteastronomy.com /encyclopedia/n/no/normal_space.htm (1278 words)

	Urysohn (Site not responding. Last check: 2007-10-09)
		After Urysohn's death, Aleksandrov argued that although Urysohn's definition of dimension was given for a metric space, it is, nevertheless, completely equivalent to the definition given by Menger for general topological spaces.
		Urysohn spotted an error in Brouwer's paper regarding a definition of dimension while he was studying it in Göttingen and easily constructed a counter-example.
		Urysohn's main contributions, in addition to the theory of dimension discussed above, are the introduction and investigation of a class of normal surfaces, metrization theorems, and an important existence theorem concerning mapping an arbitrary normed space into a
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Urysohn.html (1354 words)

	PlanetMath: separation axioms
		The separation axioms are additional conditions which may be required to a topological space in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases.
		spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.
		See Also: normal, Hausdorff space not completely Hausdorff, Sierpinski space, metric spaces are Hausdorff, zero dimensional, T2 space, regular space,
	planetmath.org /encyclopedia/SeparationAxioms.html (502 words)

	convergence spaces (Site not responding. Last check: 2007-10-09)
		Convergence spaces are for topological spaces like complex numbers are for real numbers; where some topological problems fail to find their solutions in topologies, they will, however, in convergences.
		The class of sequential topological spaces is of particular interest, on one hand because it is exactly the class of spaces for which sequences suffice to describe the topology, on the other hand because this is exactly the class of topological quotient of metrizable spaces.
		The subclass of sequential spaces that are stable under subspaces is that of Fréchet-Urysohn spaces.
	www.cs.georgiasouthern.edu /faculty/mynard_f/convergences.htm (3488 words)

	Urysohn's Lemma (Site not responding. Last check: 2007-10-09)
		Urysohn (biography) developed an equivalent criterion for a normal space.
		We can use Urysohn's lemma to build a continuous function on s that is 0 on e and 1 on f.
		A Hausdorff space s is normal iff, for every pair of disjoint nonempty closed sets e and f, there is a continuous map g(s) into [0,1] such that g(e) = 0 and g(f) = 1.
	www.mathreference.com /top,ury.html (481 words)

	PlanetMath: completely Hausdorff
		A synonym for functionally Hausdorff space is Urysohn space [1].
		For example, the term completely Hausdorff space is also used to mean a functionally Hausdorff space (e.g.
		Cross-references: implication, mean, term, closures, neighborhoods, disjoint, topological space
	planetmath.org /encyclopedia/CompletelyHausdorff.html (111 words)

	HJM, Vol. 31, No. 2, 2005
		This result includes the particular situation in which the target spaces (Y,S) are spheres of odd dimension, equipped with the standard free periodic homeomorphism of period p.
		This nonlinear connection induces a linear connection D on the total space of the tangent bundle of order k, that is called the Berwald connection.
		spaces, and deduce a characterisation of the domain under which these opersators are generators of strongly continuous semigroups.
	hjm.math.unizh.ch /Vol31-2.html (1654 words)

	PlanetMath:
		uniformly convex space is reflexive owned by georgiosl
		uniform structure (in uniform space) owned by n3o
		Urysohn function (in Urysohn's lemma) owned by yark
	planetmath.org /encyclopedia/U (716 words)

	AMCA: Discrete subspaces and topologies they determine by Vladimir Tkachuk (Site not responding. Last check: 2007-10-09)
		A] is in the closure of a discrete subspace of A. The purpose of this talk is to present some results about the classes of discretely generated and weakly discretely generated spaces which are both wider than either the class of Fréchet-Urysohn spaces or the class of scattered ones.
		We prove that the (weak) discrete generability is (closed) hereditary; every compact space is weakly discretely generated and not necessarily discretely generated.
		We prove that it is consistent with ZFC that there are countably compact Tychonoff spaces which are not weakly discretely generated.
	at.yorku.ca /c/a/e/u/13.htm (325 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		T_2.5 spaces are sometimes called completely Hausdorff spaces or Urysohn spaces.
		Subject: Re: regular 2nd countable spaces Date: Fri, 3 Aug 2001 16:10:11 -0500 Newsgroups: sci.math I am accustomed to T_2.5 meaning completely Hausdorff or Urysohn in the stronger sense, viz., for any two points p != q there is a continuous function into [0,1] with f(p) = 0 and f(q) = 1.
		(But I've also seen 'Urysohn space' used in > > a stronger sense.) > I am accustomed to T_2.5 meaning completely Hausdorff or Urysohn in the > stronger sense, viz., for any two points p != q there is a continuous > function into [0,1] with f(p) = 0 and f(q) = 1.
	www.math.niu.edu /~rusin/known-math/01_incoming/T_n (430 words)

	DC MetaData for: Random and Universal Metric Spaces (Site not responding. Last check: 2007-10-09)
		universal Urysohn space which he had defined in 1924 in his last paper.
		This means that Urysohn space is generic in the set of all Polish spaces.
		one a random Polish space is again the Urysohn space}.
	www.esi.ac.at /Preprint-shadows/esi1234.html (208 words)

	Probability Abstract Service (Site not responding. Last check: 2007-10-09)
		Vershik We continue to study the set of all metric spaces in terms of the cone of distance matrices which was suggested in the previous papers (see math.GT/0203008), and consider topological and probabilistic problems connected with this object.
		Here we prove of that the generic Polish space in the sense of this model is the so called universal Urysohn space which was defined by P.S.Urysohn in the 1920-th.
		A natural construction of a wide class of measures on the cone $\cal R$ is given and for these we show that with probability one the random Polish space is again the Urysohn space.
	www.economia.unimi.it /PAS/Letters/letter_69.shtml (5654 words)

	Professor C. Ward Henson Abstract (Site not responding. Last check: 2007-10-09)
		Urysohn's metric space U is a complete, separable metric space that (a) contains an isometric copy of each finite metric space and (b) is isometrically homogeneous for its finite subspaces.
		Given a separable metric space M, Aut(U) acts naturally on the set of isometric embeddings of M into U (by composition).
		It was shown in the 1950s that transitivity also holds when M is a compact metric space, and some examples of noncompact M were known for which this action is not transitive.
	www.math.uiuc.edu /hilda/htmlcalendars/Sep25_00/henson_sep26-00.html (155 words)

	Problem Section
		(Hattori and Ohta) A metric space is said to have UMP (resp.
		Is a space in which each closed set is R-closed [resp.
		U-closed] extension of a space X and f a continuous function from X to an R-closed [resp.
	at.yorku.ca /b/a/a/h/02.l2h (891 words)

	abstract (Site not responding. Last check: 2007-10-09)
		Metric spaces are examples of continuously Urysohn spaces with the additional property that the functions f
		We show that spaces with this property are precisely the spaces that have a weaker metric topology.
		We show that a continuously Urysohn space may fail to be strongly separating.
	iamwww.unibe.ch /~halbeis/publications/urysohn_abs.html (185 words)

	Atlas: Frechet-Urysohn topological vector spaces by Stephen A. Saxon (Site not responding. Last check: 2007-10-09)
		A topological space in which the closed subsets coincide with the sequentially closed ones is said to be Fréchet-Urysohn.
		This contradicts Observation 8.2.11(a) of the generally indispensable "Barrelled Locally Convex Spaces" by Pérez Carreras and Bonet.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cado-85.
	atlas-conferences.com /c/a/d/o/85.htm (325 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		In this paper, we show that every countably compact and countably Fréchet-Urysohn space is a sequentially compact space.
		Also we show that the Fréchet-Urysohn expansion of a countably compact and countably Fréchet-Urysohn space is a maximal sequentially compact space.
		Finally, we show that every maximal countably compact, countably Fréchet-Urysohn space is Fréchet-Urysohn.
	www.pphmj.com /abstracts/fjms/vol13issue3/AB-10.htm (79 words)

	Citebase - Kantorovich metric:initial history and little-known applications
		Authors: Vershik, A. The short history of L.Kantorovich's transport problem and his metric in the framework of his activity in mathmatics and economics during a long and difficult period.
		This is a detale version of the talk of author on the Conference devoted to 90-th annyversary of Leonid.V.Kantorovich which was at St.Petersbutrg Euler International Mathematical Institute in January 2004.
		We prove that the isometry group \Iso(\Ur) of the universal Urysohn metric space \Ur equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentratio
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0503035 (965 words)

	Atlas: Separation Pseudocharacter and Cardinality of Topological Spaces by Dimitrina N. Stavrova
		If X is Urysohn space then X aL(X, X)·U\Psi(X) In [3] A.Gryzlov and D. Stavrova proved that for a given subset X
		)·H\Psi(X) Bella A., Cammaroto F., On the cardinality of Urysohn spaces, Canad.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-41.
	atlas-conferences.com /c/a/e/u/41.htm (406 words)

	Czechoslovak Mathematical Journal, Vol. 51, No. 1, pp. 15-28, 2001 (Site not responding. Last check: 2007-10-09)
		Abstract: We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $\cgoth$ has a dense regular subspace.
		Examples are given of countable Hausdorff spaces of weight $\cgoth$ which do not have dense Urysohn subspaces.
		On the other hand, we establish that every Hausdorff space of $\oldpi$-weight less than $\gop$ has a dense completely Hausdorff (and hence Urysohn) subspace.
	cmj.math.cas.cz /cmj51-1/2.html (280 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Series: Penn State Logic Seminar Date: Tuesday, February 15, 2005 Time: 2:30 - 3:45 PM Place: 103 Pond Laboratory Speaker: John Clemens, Penn State, Mathematics Title: Fixed Points of Isometries of the Urysohn Space Abstract: A current area of research is investigating the isometry group of Urysohn's universal metric space.
		In particular, the complexity of classifying isometries up to conjugacy is unknown.
		I will show that one potential invariant turns out to be very weak in this regard, namely: Given an isometry of Urysohn's space with at least one fixed point, the set of fixed points is in fact isometric to the entire space.
	www.math.psu.edu /simpson/logic/seminar/050215.html (113 words)

	I P M - Bulletin Board (Site not responding. Last check: 2007-10-09)
		Earlier, Urysohn had constructed a Polish space (a complete separable metric space) with similar homogeneity and universality properties.
		It took longer for mathematicians to become interested in this space; but there are now some results on its isometry group.
		For example, it has an isometry all of whose orbits are dense.
	www.ipm.ac.ir /IPM/news/ViewNewsInfo.jsp?NTID=138 (390 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-10-09)
		math.GN/0509402 The isometry group of the Urysohn space as a Levy group.
		math.FA/9903085 Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space.
		funct-an/9212001 The free abelian topological group and the free locally convex space on the unit interval.
	front.math.ucdavis.edu /author/Pestov-V* (352 words)

	Outline: Topology M.S. Comprehensive Exam (Site not responding. Last check: 2007-10-09)
		Connected space, continuous image of a connected space, path connected space, continuous image of a path connected space
		Compact spaces, tube lemma, finite intersection property, closed and bounded subsets of Euclidean spaces, Cantor sets
		Regular space, completely regular space, normal space, Urysohn lemma, Tietze extension theorem
	www.math.montana.edu /Documents/Comps/ms_math/Outlines/MS_Topology (156 words)

	CiteULike: Tag urysohn (Site not responding. Last check: 2007-10-09)
		The isometry group of the Urysohn space as a Levy group
		Extension theorems and reconstruction theorems for the Urysohn Universal Space
		posted to metric-geometry metric-space random-matrix urysohn vershik by ansobol as
	www.citeulike.org /tag/urysohn (56 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-10-09)
		math.PR/0304282 Fock factorizations, and decompositions of the $L^2$ spaces over general Levy processes.
		math.GT/0203008 Distance matrices, random metrics and Urysohn space.
		math.RT/9601215 Ergodic unitarily invariant measures on the space of infinite Hermitian matrices.
	front.math.ucdavis.edu /author/Vershik-A* (272 words)

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