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Topic: Separable metric space

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	Separable space - Wikipedia, the free encyclopedia
		Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.
		The space of continuous functions on the unit interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (this is the Weierstrass approximation theorem).
		A metric space is separable iff it is second-countable and iff it is Lindelöf.
	en.wikipedia.org /wiki/Separable_space (710 words)

	separable (topology) (Site not responding. Last check: 2007-10-16)
		Separable spaces are therefore topological spaces with a certain limit on their size: an uncountable discrete space isn't separable.
		Most of the spaces initially encountered are indeed separable: for example the real numbers with their standard metric have the rational numbers as a countable dense subset.
		Since the space of continuous functions on the interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (see Weierstrass approximation theorem), and their coefficients can be approximated by rationals, that space is also separable.
	www.yourencyclopedia.net /Separable_(topology).html (397 words)

	Separable space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-16)
		For example taking Hilbert space to mean a complex (A metric space that is linear and complete and (usually) infinite-dimensional) Hilbert space of infinite dimension and separable, there is one such space up to isomorphism (there is a categorical theory, at least if our theory of the real numbers is categorical).
		Separability is especially important in (Click link for more info and facts about numerical analysis) numerical analysis and (Click link for more info and facts about constructive mathematics) constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces.
		Also every subspace of a separable (A set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality) metric space is separable.
	www.absoluteastronomy.com /encyclopedia/S/Se/Separable_space.htm (975 words)

	Separable space (Site not responding. Last check: 2007-10-16)
		In mathematics a metric space (or topological space) X is separable if it has a countable subset Y such that members of Y any x in X as closely as we Formally a topological space is separable if and only if it has a subset that is countable and dense.
		Most of the spaces initially encountered are separable: for example the real numbers with their standard metric have the rational numbers as a countable dense subset.
		Since space of continuous functions on the interval [0 1] with the metric of uniform convergence has a dense subset of polynomials (see Weierstrass approximation theorem) and their coefficients can be approximated rationals that space is also separable.
	www.freeglossary.com /Separable_metric_space (694 words)

	metric space
		In mathematics, a metric space is a set (or "space") where a distance between points is defined.
		An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem).
		Every such metric can be rescaled to a finite metric (using d'(x, y) = d(x, y) / (1 + d(x, y)) or d''(x, y) = min(1, d(x, y))) and the two concepts of metric space are therefore equivalent as far as notions of topology (such as continuity or convergence) are concerned.
	www.fact-library.com /metric_space.html (1251 words)

	Separable (Site not responding. Last check: 2007-10-16)
		In mathematics, a metricspace (or topological space) X is separableif it has a countable subset Y suchthat members of Y approximate any x in X as closely as we like.
		Separable spaces are therefore topological spaces with a certainlimit on their size: an uncountable discrete space isn't separable.
		Separability is especially important in numerical analysis and constructive mathematics, since many theoremsthat can be proved for nonseparable spaces have constructive proofs only for separable spaces.
	www.therfcc.org /separable-76313.html (348 words)

	Reverse mathematics - Wikipedia, the free encyclopedia
		The Baire category theorem in a complete separable metric space (the separability condition is necessary to even code the theorem in the language of second order arithmetic).
		The Heine-Borel theorem for complete totally bounded separable metric spaces (where covering is by a sequence of open balls).
		The separable Hahn-Banach theorem in the form: a bounded linear form on a subspace of a separable Banach space extends to a bounded linear form on the whole space.
	www.wikipedia.org /wiki/Reverse_mathematics (3403 words)

	Real number - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-16)
		More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section).
		The reals are a contractible (hence connected and simply connected), locally compact separable metric space, of dimension 1, and are everywhere dense.
		Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra.
	encyclopedia.worldsearch.com /real_number.htm (2144 words)

	Separable space (Site not responding. Last check: 2007-10-16)
		Separable spaces are therefore topological spaces a certain limit on their size: an discrete space isn't separable.
		Separability is especially important in numerical analysis and constructive mathematics since many theorems that can be for nonseparable spaces have constructive proofs only separable spaces.
		More generally separable uniform space whose uniformity has a countable basis be second countable.
	www.freeglossary.com /Separable_space (694 words)

	Dictionary of Meaning www.mauspfeil.net
		space''', is a space in which any two distinct points can be separated by closed neighborhoods.
		This page and completely Hausdorff space currently describe the same concept: ''a space* in which points can be separated by a function'' and we have no page describing ''a space* in which points can be separated by closed neighborhoods.'' Different authors seem to have different views on which name goes with which.
		AFAICR, my only reason for picking the definitions that I did on Separation axiom is that Willard is generally more modern in tone than Steen and Seebach (despite the coincidence of their dates), but further references may (or may not) confirm that Willard's terminology really is more modern.
	www.mauspfeil.net /Completely_Hausdorff_space.html (1382 words)

	Space of Ends - Equivalence of Different Definitions (Site not responding. Last check: 2007-10-16)
		In [1] is given a description of the space of ends of a topological space by use of admissible sequences.
		This description is equivalent with the general notion in the case when the topological space is locally compact, separable metric space, and its space of quasicomponents is compact.
		In this paper we give a straight proof that the previous description of the space of ends of a connected, locally compact, metric space by use of admissible sequences, coincides with the description which uses inverse limit of the components of complements of compacta.
	www.pmf.ukim.edu.mk /mathematics/vasilevska.htm (212 words)

	Real number (Site not responding. Last check: 2007-10-16)
		More technically, the reals are complete (in the sense of metric space s or uniform space s, which is a different sense than the Dedekind completeness of the order in the previous section).
		The real numbers form a metric space : the distance between x and y is defined to be the absolute value
		Self-adjoint operator s on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra.
	www.nebulasearch.com /encyclopedia/article/Real_number.html (2265 words)

	Separable metric space (Site not responding. Last check: 2007-10-16)
		Since the space of continuous functions on the interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (see Weierstrass approximationtheorem), and their coefficients can be approximated by rationals, that space is also separable.
		For technical reasons the foundations of general topology arewritten without the requirement of separability, or other 'axioms of countability'.
		Such constructive proofs can beturned into algorithms for use in numerical analysis, and they are the only sortsof proofs acceptable in constructive analysis.
	www.therfcc.org /separable-metric-space-195559.html (348 words)

	54E: Spaces with richer structures especially metric spaces
		A set X with a metric on it is called a metric space since the collection of unions of balls is a topology on X. A major theme in research is to investigate the influence a metric has on the underlying topology.
		Among those mentioned in the MSC we observe semimetric spaces (topological spaces whose topology is given by the balls with respect to a semimetric -- a distance function not meeting the triangle inequality); cosmic spaces (continuous image of a separable metric space), and probabilistic metric spaces.
		The theory of metric spaces is almost always presented with an eye towards its connections either with general topology or with analysis; this is true both at the beginning undergraduate level and at advanced levels.
	www.math.niu.edu /~rusin/known-math/index/54EXX.html (1288 words)

	PlanetMath: Polish space
		A Polish space is a topological space that is homeomorphic to some complete separable metric space.
		Cross-references: metric space, separable, complete, homeomorphic, topological space
		This is version 1 of Polish space, born on 2005-01-31.
	planetmath.org /encyclopedia/PolishSpace.html (53 words)

	Clearing up the market cycle... best Universal Metric Space (Site not responding. Last check: 2007-10-16)
		Random Metric Spaces and Universality Random Metric Spaces and Universality WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space.
		Random and universal metric spaces Random and universal metric spaces We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems...
		Metric Space -- from MathWorld Metric Space -- from MathWorld A set S with a global distance function (the metric g) which, for every two points x, y in S, gives the distance between them as a nonnegative real number g(x,y).
	ascot.pl /th/Fourier5/Universal-Metric-Space.htm (759 words)

	1968
		This paper is concerned with relationship of this metric to the metric on the random variables that metrizes the convergence in probability.
		From this Dudley derives the equivalence of this and the Prohorov metrics.
		Radner models uncertainty as a sequence of increasingly fine partitions of a finite space.
	www.io.com /~slava/history/1968.htm (624 words)

	Quantum Theory: von Neumann vs. Dirac
		The trace of the identity is infinite when the separable Hilbert space is infinite-dimensional, which means that it is not possible to define a correctly normalized a priori probability for the outcome of an experiment (i.e., a measurement of an observable).
		In distribution theory, the space Φ is characterized as a test-function space, where a test-function is thought of as a very well-behaved function (being continuous, n-times differentiable, having a bounded domain or at least dropping off exponentially beyond some finite range, etc).
		First, dual pairs of this sort can also be generated from a pre-Hilbert space, which is a space that has all the features of a Hilbert space except that it is not complete, and doing so has the distinct advantage of avoiding the partitioning of functions into equivalence classes (in the case of functions spaces).
	setis.library.usyd.edu.au /stanford/entries/qt-nvd (10284 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Hilbert space is a separable metric space; being infinite-dimensional, it is not embeddable in any R^d with d finite; it is homeomorphic to R^omega.
		Every separable metric space is embeddable in Hilbert space.
		Every finite-dimensional separable metric space is embeddable in some R^d; in fact, every m-dimensional separable metric space is embeddable in R^{2m+1}.
	www.math.niu.edu /~rusin/known-math/01_incoming/embed_metric (145 words)

	Separable space
		A subspace of a separable space need not be separable, but every open subspace of a separable space is separable.
		This page was last modified 20:48, 15 May 2005.
		The article about Separable space contains information related to Separable space, Examples and Properties.
	www.arikah.net /encyclopedia/Separability (698 words)

	John's Research Blog: Revelation Principle--Relation to Other Literatures
		The reason is that the space of all compact (I think) preference relations is metrizable.
		That means that it is a complete, separable metric space.
		Hence, it is Borel equivalent to any other complete, separable metric space and that's good news for us.
	johnsresearch.blogspot.com /2004/06/revelation-principle-relation-to-other.html (206 words)

	Selected Materials from Talks by Andrej Bauer
		There are several generalizations of topological spaces which are "imaginary" in the sense of Martin Escardo.
		The first one states that Cantor space embeds in any inhabited complete separable metric space without isolated points, X, in such a way that every sequentially continuous function f from Cantor space to Z extends to a sequentially continuous function F from X to R.
		Second, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable.
	www.andrej.com /talks (1117 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		A separable metric space is one with a countable dense subset.
		We should be appreciative of _any_ characterization which looks at a space just topologically and says, "yup, there is a metric somewhere which defines this topology".
		I believe you also need to assume that the space is T1, although I don't have an counterexample offhand.
	www.math.niu.edu /~rusin/papers/known-math/95/metrization (283 words)

	Vasco Brattka's Papers (Site not responding. Last check: 2007-10-16)
		In formal analogy to separable metric spaces we introduce the concept of a generated quasi-metric space.
		In a corresponding way as each point of a separable metric space can be represented as the limit of a sequence in some countable dense subset, each point of a generated quasi-metric space can be considered as the infimum of a sequence in the generating set (w.r.t.
		Typically, the generating subset can be chosen such that it is itself a separable metric space (w.r.t.
	www.informatik.fernuni-hagen.de /thi1/vasco.brattka/publications/generated.html (127 words)

	Citations: When Scott is weak on the top - Edalat (ResearchIndex)
		assuming a separable metric space that embedded as a G subset of a countably based domain.
		Moreover, they show that the model is a continuous poset whose completion (as described in Section 2) XV has the completion of the metric space as its space of maximal elements.
		....space is R integrable with respect to a bounded Borel measure if and only if its set of discontinuities has measure zero and that if the function is R integrable then it is also Lebesgue integrable and the two integrals coincide.
	citeseer.ist.psu.edu /context/225597/182105 (1291 words)

	Professor C. Ward Henson Abstract (Site not responding. Last check: 2007-10-16)
		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/Oct02_00/henson_oct03-00.html (155 words)

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