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Topic: Pseudometric space

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	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.
		Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space.
		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)

	Baire space - Wikipedia, the free encyclopedia
		Ignoring spaces with isolated points, which are their own interior, a Baire space is "large" in the sense that it cannot be constructed as a countable union of its points.
		Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons.
		Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval [0, 1].
	www.wikipedia.org /wiki/Baire_space (732 words)

	Station Information - Baire space
		Every space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space (this includes the irrational numbers with their standard topology, as well as the Cantor set).
		Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletonss.
		In particular, every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval [0,1].
	www.stationinformation.com /encyclopedia/b/ba/baire_space.html (532 words)

	Metric space (Site not responding. Last check: 2007-10-22)
		Since metric spaces are topological spaces, one has a notion of continuous function between metric spaces.
		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).
		The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace.
	www.sciencedaily.com /encyclopedia/metric_space (1687 words)

	Encyclopedia: Pseudometric (Site not responding. Last check: 2007-10-22)
		A useful characterisation of compactness for metric spaces is that a metric space is compact if and only if it is complete and totally bounded.
		If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as
		For instance, the completion of a metric space M involves an isometry from M into M', a quotient of the space Cauchy sequences on M.
	www.nationmaster.com /encyclopedia/Pseudometric (1684 words)

	PlanetMath: example of pseudometric space (Site not responding. Last check: 2007-10-22)
		Note, however, that this is not an example of a metric space, since we can have two distinct points that are distance 0 from each other, e.g.
		"example of pseudometric space" is owned by mathcam.
		This is version 2 of example of pseudometric space, born on 2004-10-02, modified 2004-10-03.
	planetmath.org /encyclopedia/TrivialPseudometric.html (110 words)

	Station Information - Tychonoff space
		In topology and related brances of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces.
		Tychonoff spaces are named after Andrey Tychonoff, whose Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov".
		Every locally compact regular space is completely regular, and every locally compact Hausdorff space is Tychonoff.
	www.stationinformation.com /encyclopedia/t/ty/tychonoff_space.html (409 words)

	Metric space : Pseudometric (Site not responding. Last check: 2007-10-22)
		A metric space is a space where a distance between points is defined.
		A metric space in which every Cauchy sequence has a limit is said to be complete.
		A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely).
	www.termsdefined.net /ps/pseudometric.html (1312 words)

	Tychonoff space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		Tychonoff spaces are named after (Click link for more info and facts about Andrey Nikolayevich Tychonoff) Andrey Nikolayevich Tychonoff, whose (A native or inhabitant of Russia) Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov".
		Every (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 Tychonoff; every (Click link for more info and facts about pseudometric space) pseudometric space is completely regular.
		More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K and an (Click link for more info and facts about injective) injective continuous map j from X to K such that the (Something inverted in sequence or character or effect) inverse of j is also continuous.
	www.absoluteastronomy.com /encyclopedia/T/Ty/Tychonoff_space.htm (873 words)

	Baire space: Definition and Links by Encyclopedian.com - All about Baire space
		In topology, a Baire space is a particular type of topological space in which, intuitively, there are "enough" points for certain limit processes.
		every space that is homeomorphic to an open subset of a complete pseudometric space is a Baire space (this includes the irrational numbers)
		Note that the space of rational numbers with their ordinary topology are not a Baire space, since they are the union of countably many nowhere dense sets, the singletons.
	www.encyclopedian.com /ba/Baire-space.html (282 words)

	Clearing up the market cycle... best Pseudometric (Site not responding. Last check: 2007-10-22)
		Pseudometric -- from MathWorld Pseudometric -- from MathWorld A distance g on a set that fulfils the same properties as a metric except relaxes the definition to allow the distance between two different points to be zero.
		The Kobayashi pseudometric on algebraic manifold and a canonical fibration
		The Kobayashi pseudometric on algebraic manifold and a canonical fibration The Kobayashi pseudometric on algebraic manifold and a canonical fibration Given a compact complex manifold X of dimension n, we define a bimeromorphic invariant...
	ascot.pl /th/Fourier5/Pseudometric.htm (553 words)

	Baire space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		In a topological space we can think of (Click link for more info and facts about closed set) closed sets with (A container that has been emptied) empty (The region that is inside of something) interior as points in the space.
		As a topological space, B is (Click link for more info and facts about homeomorphic) homeomorphic to the set Ir of (A real number that cannot be expressed as a rational number) irrational numbers carrying their standard topology inherited from the reals.
		Baire space should be contrasted with (Click link for more info and facts about Cantor space) Cantor space, the set of infinite sequences of (Either 0 or 1 in binary notation) binary digits.
	www.absoluteastronomy.com /encyclopedia/B/Ba/Baire_space.htm (1053 words)

	Metric space : Pseudometric space (Site not responding. Last check: 2007-10-22)
		Every metric space is isometrically isomorphic to a closed subset of some normed vector space.
		Every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
		If one drops property 3, one obtains pseudometric spaces; if one drops property 4 instead, one obtains quasimetric spaces[?].
	www.eurofreehost.com /ps/Pseudometric_space_4.html (330 words)

	PlanetMath: pseudometric space (Site not responding. Last check: 2007-10-22)
		In other words, a pseudometric space is a generalization of a metric space in which we allow the possibility that
		See Also: metric space, quasimetric space, normed vector space, seminorm
		This is version 4 of pseudometric space, born on 2004-10-02, modified 2004-10-07.
	planetmath.org /encyclopedia/Pseudometric.html (74 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Linear topological spaces with pseudometric which satisfies the time inequality instead of the triangle inequality are studied (3 axioms).
		Pseudometric (which is determined by a pseudonorm) is shown to define a topology on a linear space, it being a continuous mapping in this topology.
		Proved that for a space with pseudometric to be a linear kinematics it is necessary and sufficient that mapping of multiplication by -1 (i.e.
	www.cs.odu.edu /~dlibug/ups/rdf/xxx/gr-qc/9704016.rdf (208 words)

	Clearing up the market cycle... best Pseudometric Topology (Site not responding. Last check: 2007-10-22)
		Pseudometric Topology -- from MathWorld Pseudometric Topology -- from MathWorld A topology on a set X whose open sets are the unions of open balls B(X_0,r)=\{x\in x\mid g(x_0,x) 0.
		It's customary to treat the relativistic spacetime manifold as an ordinary topological space with the same topology as a four-dimensional Euclidean manifold, denoted by R4.
		In a pseudometric space, a point p is in the closure of...
	ascot.pl /th/Fourier5/Pseudometric-Topology.htm (510 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		In this formalism, we consider the difference between maps to be the distance between elements of a pseudometric space (the space of all such maps).
		Implicit in the literature on these two mechanisms is that the parameter space available for binary formation in a star forming region varies with the cloud temperature.
		For each physical characteristic of interest, this formal system assigns a distance function (a pseudometric) to the space of all maps; this procedure allows us to measure quantitatively the difference between any two maps and to order the space of all maps.
	www.ifa.hawaii.edu /~reipurth/newsletter/newsletter22.tex (4511 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Every closed filter in a $T_0$ topological space can be extended to a maximal closed filter.
		Every non-empty complete pseudometric space with a countable base is of the second category (non-meager).
		Every non-empty complete totally bounded pseudometric space is of the second category (non-meager).
	www.math.purdue.edu /~jer/cgi-bin/new-forms.tex (1811 words)

	Baire Category Theorem Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-22)
		The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.
		Note that neither of these statements implies the other, since there is a complete metric space which is not locally compact (the Baire space of irrational numbers), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space).
		Relation to AC The proofs of BCT1 and BCT2 require some form of the axiom of choice; and in fact the statement that every complete pseudometric space is a Baire space is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice.
	www.karr.net /encyclopedia/Baire_category_theorem (576 words)

	Topics: Distances and Metric Spaces
		Pseudometric space: A pair (X,d) with d a pseudodistance on X.
		Baire category theorem: A complete metric space is not the countable union of nowhere dense sets; Can be stated as a theorem in ramsey theory.
		Quantum metric space: A C-algebra (or more generally an order-unit space*) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric.
	www.phy.olemiss.edu /~luca/Topics/d/distance.html (449 words)

	The Ultimate Talk:Kolmogorov space Dog Breeds Information Guide and Reference (Site not responding. Last check: 2007-10-22)
		In this view, one could define metrizable topological spaces as those that come from a metric, and pseudo-metrizable topological spaces as those whose KQ comes from a metric.
		But placing structures on a topological space is more general (or can be, depending on how you define structure), and almost any structure used in practice can be transferred this way, whether it uniquely defines the topology or not.
		And saying that the KQ of a pseudometric space is a metric space is certainly saying more than that the KQ of a pseudometrisable space is a metrisable space; the KQ doesn't just give you the property of being pseudometrisable but the structure of having a specific pseudometric.
	www.dogluvers.com /dog_breeds/Talk:Kolmogorov_space (296 words)

	Kunzi (Site not responding. Last check: 2007-10-22)
		A quasi-uniform space X is hereditarily precompact if and only if each ultrafilter on X is a left K- Canchy filter.
		A quasi-uniform space is Smyth completable if and only if each left K-Cauchy filter is stable.
		In a quasi-pseudometric space (X, d) sequential left K-completeness, left K-completeness and the property that each filter which is co-stable (that is, stable in the conjugate) clusters in (X, d) are equivalent.
	www.unina2.it /topological.sun/Kunzi.html (189 words)

	HAF: Mathematical Errors (Site not responding. Last check: 2007-10-22)
		Then a sequence of zero functions converges in pseudometric to the characteristic function of N even though 0 is strongly measurable and the characteristic function of N is not.
		For instance, if the measure space is [0,1] with Lebesgue measure, let N be an uncountable set with measure 0, and let f map the points of N to a non-separable range; then f is not strongly measurable but f is equivalent to strongly measurable functions.)
		In the proof that (B) implies (C) and (E) in normed spaces, the last part of the argument is missing a few steps.
	www.math.vanderbilt.edu /~schectex/ccc/addenda/matherr.html (1774 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Let X and Y be complete metric spaces and f_n : X --> Y continuous, with f_n --> f pointwise.
		For example, if T_n is a continuous linear operator from one Banach space X to another, and T_n converges pointwise to T, then T must be continuous at SOME point, hence at ALL points; this is the most-frequently-used consequence of the Uniform Boundedness Principle.
		nu implies their continuity on this metric space; hence the limit, mu, is continuous at SOME point; continuity at SOME point implies continuity at the empty set; which in turn implies continuity at all points; which in turn implies mu is a measure.
	www.math.niu.edu /~rusin/known-math/01_incoming/vitali-hahn-saks (491 words)

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