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# Topic: Hausdorff space

###### In the News (Tue 18 Jun 19)

 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. In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. en.wikipedia.org /wiki/Hausdorff_space   (1227 words)

 Completely Hausdorff space - Wikipedia, the free encyclopedia In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space. space, is a space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a space in which any two distinct points can be separated by a function. en.wikipedia.org /wiki/Completely_Hausdorff_space   (442 words)

 Locally compact space - Wikipedia, the free encyclopedia Almost all locally compact spaces studied in applications are Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff spaces. Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article Compact space. All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology. en.wikipedia.org /wiki/Locally_compact_space   (1326 words)

 Compact space   (Site not responding. Last check: 2007-10-22) In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R 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. hallencyclopedia.com /Compact_space   (1635 words)

 Hausdorff space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-22) space is a ((mathematics) any set of points that satisfy a set of postulates of some kind) topological space in which points can be separated by neighbourhoods. The Hausdorff condition is one in a series of (Click link for more info and facts about separation axiom) separation axioms that can be imposed on a topological space, however it is the one that is most frequently used and discussed. Almost all spaces encountered in (The abstract separation of a whole into its constituent parts in order to study the parts and their relations) analysis are Hausdorff; most importantly, the (Any rational or irrational number) real numbers are a Hausdorff space. www.absoluteastronomy.com /encyclopedia/H/Ha/Hausdorff_space.htm   (1626 words)

 Encyclopedia: Completely Hausdorff space Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. www.nationmaster.com /encyclopedia/Completely-Hausdorff-space   (1117 words)

 PlanetMath: T2 space Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces. This is version 15 of T2 space, born on 2002-02-08, modified 2005-05-10. Of cource if he is determined enough, he could look through every entry with the name Hausdorff and finally finds what he needs, but it is better to have a direct link with name "Hausdorff space". planetmath.org /encyclopedia/HausdorffTopology.html   (290 words)

 [No title] Baire space +------------------------------------------------------------ A Baire space is a topological space with the property that the intersection of countable family of open dense subsets is dense. Hausdorff +------------------------------------------------------------ A topological space (X,T) is called Hausdorff if for every two points x,y in X, there are disjoint open sets U,V in T such that x in U and y in V. This is refined through seperation axioms, T0,..., T4. Loops play a role in definitions like simply connected: a topological space is simply connected if every loop is homotopic to a constant loop which is a fancy way telling that every closed path can be collapsed inside X to a point. www.math.harvard.edu /~knill/sofia/data/topology.txt   (1652 words)

 Hausdorff space   (Site not responding. Last check: 2007-10-22) space, or separated space, iff, given any distinct points x and y, there are a neighbourhood U of x and a neighbourhood V of y that are disjoint. Similarly, a space is preregular iff all of the limits of a given net (or filter) are topologically indistinguishable. In fact, a quotient space of a Hausdorff space X is itself Hausdorff if and only if the kernel of the quotient map is closed as a subset of the Cartesian product X × X. www.sciencedaily.com /encyclopedia/hausdorff_space   (881 words)

 PlanetMath: Hausdorff space not completely Hausdorff We will use the closed-neighborhood sense for completely Hausdorff, which will also imply the topology is not completely Hausdorff in the functional sense. "Hausdorff space not completely Hausdorff" is owned by drini. This is version 18 of Hausdorff space not completely Hausdorff, born on 2004-03-17, modified 2005-05-10. planetmath.org /encyclopedia/HausdorffSpaceNotCompletelyHausdorff.html   (326 words)

 Normal space X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E and a neighbourhood V of F that are also disjoint. An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry. A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. usapedia.com /n/normal-space.html   (914 words)

 Metric space - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-22) In metric spaces, one can talk about limits of sequences; 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). 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). xahlee.org /_p/wiki/Metric_spaces.html   (1390 words)

 Hausdorff space Article, Hausdorffspace Information   (Site not responding. Last check: 2007-10-22) Pseudometric spaces typically arenot Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry,in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. Compact preregular spaces are normal, meaning that they satisfy Urysohn'slemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers. www.anoca.org /spaces/preregular/hausdorff_space.html   (825 words)

 [No title]   (Site not responding. Last check: 2007-10-22) In fact, in a normal space, any two disjoint sets will also be separated by a function; this is Urysohn's Lemma. Note that a normal Hausdorff space must also be both Tychonoff and normal regular. There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. www.online-encyclopedia.info /encyclopedia/s/se/separation_axiom.html   (1450 words)

 Hausdorff space - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-22) In topology and related branches of mathematics, Hausdorff spaces and preregular spaces are kinds of topological spaces. Limits of sequences, nets, and filters (when they exist) are unique in Hausdorff spaces. A topological space X is Hausdorff if and only if the diagonal {(x,x) : x in X} is a closed set in X × X, the Cartesian product of X with itself. xahlee.org /_p/wiki/Hausdorff_space.html   (828 words)

 A homotopy double groupoid of a Hausdorff space   (Site not responding. Last check: 2007-10-22) A homotopy double groupoid of a Hausdorff space We associate to a Hausdorff space, $X$, a double groupoid, $\mbox{\boldmath$ \rho $}^{\square}_{2} (X)$, the homotopy double groupoid of $X$. The construction is based on the geometric notion of thin square. www.tac.mta.ca /tac/volumes/10/2/10-02abs.html   (165 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)

 PlanetMath: proof of A compact set in a Hausdorff space is closed PlanetMath: proof of A compact set in a Hausdorff space is closed "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   (161 words)

 [No title] If X is a first countable non-Hausdorff space, then there is a sequence which converges to at least two points. If X is a non-Hausdorff space, then there is a directed net which converges to at least two points. It would be interesting to have an example of a non-Hausdorff space in which every sequence, whatever the lenght, converges to at most one point. www.math.niu.edu /~rusin/known-math/99/nonhausdorff   (536 words)

 Gravity: Introduction   (Site not responding. Last check: 2007-10-22) Hausdorff space, in which this statement can be made. It is a major theorem that every Hausdorff space is a metric space, and every metric space is a Riemannian manifold (and conversely). With the resulting topology the space is not a Hausdorff space, not a metric space and not a Riemannian manifold; it is junk, and physical results can only be recovered by ignoring the topology induced by the Minkowski metric. www.math.ucla.edu /~jimc/klein_h/intro.html   (357 words)

 Hausdorff and Other Separation Axioms for Topological Spaces   (Site not responding. Last check: 2007-10-22) The fully general, abstract topological spaces (S,T), where S is a set and T is a collection of subsets of S that is closed under arbitrary unions and finite intersections and includes both the null set ∅ and the whole set S, is of limited interest. (Hausdorff Space Axiom): For any two points p and q in the topological space (S,T) there is a pair of disjoint open sets, one containing {p} and not containgin {q} and the other containing {q} and not {p}. (Regular Space Axiom): For any closed set C in the topological space (S,T) and any point p not in C there is a pair of disjoint open sets, one containing C and one containing {p}. www.applet-magic.com /separation.htm   (285 words)

 Separation axioms In a Hausdorff space, distinct points are "housed off" from one another by open sets. It follows that every finite set is closed in a Hausdorff space and the topology is therefore stronger than the cofinite topology. space with a countable basis is metrisable (that is, the topology may be obtained from a metric). www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L18.html   (481 words)

 Gravity: Notation and Definitions A topological space in which, for every pair of nonidentical points p and q, there exist two open sets P and Q containing them, that do not intersect (that is, whose intersection is empty). It is a major theorem that every Hausdorff space is a metric space, that is, it has a distance function (and vice versa, an easy theorem). A manifold is a Hausdorff space, with a countable basis of open sets, such that a neighborhood of every point (some open set containing the point) is homeomorphic to R www.math.ucla.edu /~jimc/klein_h/notation.html   (1120 words)

 Selections and suborderability by G. Artico, U. Marconi, J.Pelant, L. Rotter, and M. Tkachenko   (Site not responding. Last check: 2007-10-22) A selection on an Hausdorff space X is a Vietoris continuous selection on all non-empty closed subsets of X. A weak selection is a continuous selection on all subsets consisting of one or two points. Furthermore, a sequentially compact suborderable space is not necessarily orderable. There exists a locally compact locally countable space with a zero-selection and Cantor-Bendixson height equal to 2 which is not normal. at.yorku.ca /i/a/a/i/09.htm   (509 words)

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