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Metrization theorem - Wikipedia, the free encyclopedia Urysohn had shown, in a paper published posthumously in 1925, was the slightly weaker result that every second-countable normal Hausdorff space is metrizable.) uniformizability, a topological space homeomorphic to a uniform space sufficient conditions for a topological space to be metrizable. en.wikipedia.org /wiki/Metrization_theorems

Discrete space Information - TextSheet.com The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense. However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by one and nonexpansive maps. www.shopping.top5miami.com /encyclopedia/d/di/discrete_space.html

PlanetMath: second countable See Also: separable, Lindelöf, every second countable space is separable, Lindelöf theorem, Urysohn metrization theorem, first axiom of countability A topological space is said to be second countable if it has a countable basis. It can be shown that a second countable space is both Lindelöf and separable, although the converses fail. www.planetmath.org /encyclopedia/SecondAxiomOfCountability.html

First-countable space - Wikipedia, the free encyclopedia An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line). In first-countable spaces, sequential compactness and countable compactness are equivalent properties. Specifically, a space X is said to be first-countable if each point has a countable local base. en.wikipedia.org /wiki/First_countable

Topological space - Wikipedia, the free encyclopedia Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods. A linear graph is a topological space that generalises many of the geometric aspects of graphs with vertices and edges. en.wikipedia.org /wiki/Topological_space

Discrete space Information - TextSheet.com Every discrete space is first countable, and a discrete space is second countable iff it is countable. The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense. www.shopping.top5miami.com /encyclopedia/d/di/discrete_space.html

Paracompact space Information - TextSheet.com For instance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is then extended to the whole space via a partition of unity. A paracompact space is a topological space in which every open cover admits an open locally finite refinement. The most important feature of paracompact Hausdorff spaces is that they are normal and admit partitions of unity relative to any open cover. forum.top5miami.com /encyclopedia/p/pa/paracompact_space.html

Second-countable space - Wikipedia, the free encyclopedia For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Although the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates. en.wikipedia.org /wiki/Second_countable

Separable space - Art History Online Reference and Guide A metric space is separable iff it is second-countable and iff it is Lindelöf. 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). This condition is typical of spaces that are met in classical parts of mathematical analysis and geometry. www.arthistoryclub.com /art_history/Separable_space

Second-countable space -- Facts, Info, and Encyclopedia article In second-countable spaces—as in metric spaces— (The consistency of a compact solid) compactness, sequential compactness, and countable compactness are all equivalent properties. Specifically, a space is said to be second-countable if its topology has a (Click link for more info and facts about countable) countable (Installation from which a military force initiates operations) base. In (The configuration of a communication network) topology, a second-countable space is a ((mathematics) any set of points that satisfy a set of postulates of some kind) topological space satisfying the "second (Click link for more info and facts about axiom of countability) axiom of countability ". www.absoluteastronomy.com /encyclopedia/S/Se/Second-countable_space.htm

Profile of the Faculty The most common objects in topology, such as manifolds, polyhedra, and Euclidean spaces, are metrizable spaces, which means that one can describe the topology by a (real-valued) distance between points, and say that a sequence of points converges to a limit if and only if the distance decreases to zero. In particular, he is interested in stratifiable spaces and other spaces which have some but not all properties of metrizable spaces. The study of these general spaces is called general topology and it has connections to analysis and logic. www.math.ucdavis.edu /research/profiles/cborges

1st_countable In fact, Leslie Foged constructed 2^c nonhomeomorphic countable spaces having no points of first countability in his 1979 Ph.D. Dissertation "Weak Bases for Topological Spaces" under Ron Freiwald at Washington University. As for what you want to prove, namely that for countable topological spaces, first countability implies second countability, you want to be a bit more explicit in your proof. First, tell your reader exactly which countable collection of open sets you're going to show is a base. www.math.niu.edu /~rusin/known-math/01_incoming/1st_countable

math.html Malykhin \cite{Malykhin} used this method to build (again under the continuum hypothesis) a space of cardinality $\gc^+$ that is Hausdorff, compact, and weakly first-countable. Jakovlev \cite{Jakovlev} constructed under the continuum hypothesis a compact Hausdorff space of cardinality $\gc$ which is weakly first-countable but not first-countable. A combinatorial statement concerning ideals of countable subsets of $\omega_1$ is introduced and proved to be consistent with the Continuum Hypothesis. www.cs.bgu.ac.il /~abraham/math.html

Articles - Closed set In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets. 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. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. www.free-biz.org /articles/Closed

First and Second Countable If every point p in a topological space has a countable base at p, the base is first countable. This is an indispensable property of first countable, and it is used in various proofs. The most common example is a metric space, where base sets are open balls, and each base set is assigned to its center. www.mathreference.com /top,12cnt.html

Surface Surfaces are tangible in three-dimensional space only as the boundaries of three-dimensional solid objects. In mathematics, a surface is a two-dimensional manifold. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. www.brainyencyclopedia.com /encyclopedia/s/su/surface.html

Selections and suborderability by G. Artico, U. Marconi, J.Pelant, L. Rotter, and M. Tkachenko There exists a locally compact locally countable space with a zero-selection and Cantor-Bendixson height equal to 2 which is not normal. There exists a locally compact locally countable space which is hereditarily collectionwise normal and has a zero-selection, but it is not suborderable. 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. at.yorku.ca /i/a/a/i/09.htm

mc25 Clearly Q and N are first countable as are Q times Q and Q times N. Generally, an arbitrary quotient space of a first countable space need not be first countable. Similarly, Q times N is first countable as are (Q smash Q) times N and Q times (Q smash N). There is a well known and easily proved fact that spaces that satisfy the first axiom of countability are k spaces. www.lehigh.edu /~dmd1/mc25

m425 Consider Y = [0, 2] as a subspace of R. Then [0, 1) is open in Y but not open in R. A first countable space is always second countable. Topics include metric spaces, topological spaces, continuous functions, connectedness, compactness, countability and separation axioms, the fundamental group, and the classification of surfaces. A closed subset of a Hausdorff space is compact. ac.marywood.edu /johnsonc/www/m425.htm

PlanetMath: $\mathcal{C}^r$ topologies Cross-references: Baire space, compact, first countable, separable, metric, complete, metrizable, covering, finite, compact-open topology, basis, local coordinates, sequence, charts, compact sets, atlas, locally finite, neighborhood, embedding, properties, derivatives, continuous, manifold, mappings, topology is not even first countable (thus, not metrizable) when Whitney (or strong) topology is a topology assigned to the space planetmath.org /encyclopedia/CompactOpenMathcalCrTopology.html

topo101 On the other hand, you can't expect the stronger results to apply in general, and you may have a hard time coming up with "internal" properties of a topological space (Hausdorff, first countable, etc.) which would allow you to draw the desired conclusions. This is how some of the properties of topological spaces first get enunciated: one looks to see which properties [0,1] has, and which of those properties are really necessary to make some proof work. CW complexes, manifolds, metric spaces,...) you can get stronger results which are more directly applicable to certain situations which arise outside Topology. www.math.niu.edu /~rusin/known-math/99/topo101

(at).(at) topology~ If the space isn't first countable, there can be a sequence which has an accumulation point but no convergent subsequence. 1st countable T1 space where all converging sequences are eventually constant is discrete, hence anti-compact. Turns out Arens space is Hausdorff and anti-compact, ie all compact sets are finite. www.forum-one.org /new-6333553-4346.html

Long line (topology) - Enpsychlopedia L * is not a manifold and is not first countable. A related space, the extended long line, L *, is obtained by adjoining an additional element to the end of L. In topology, the long line is a topological space analogous to the real line, but much longer. www.grohol.com /psypsych/Long_line_(topology)

research The main result is that the fundamental group of a second countable, connected, locally path connected one-dimensional metric space is free if and only if it is countable, if and only if the space has a universal cover. We use these equivalencies to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. To begin, we define several homotopy theoretic conditions which we then prove are equivalent to the existence of a universal covering space. www.math.byu.edu /~conner/research

Glossary of research economics in a region of a parameter space the power of a test goes to one as sample size n goes to infinity, that test is said to be consistent against alternatives in that region of the parameter space. countable additivity property: the third of the properties of a measure. It can be constructed by a countable number of non-overlapping open rectangles (since a series of such rectangles can be defined that would cover every point in the circle but no point outside of it. www.econterms.com /econtent.html

Nonlinear Science FAQ If we have such a Baire space of dynamical systems, and there is a property which is true on a countable intersection of open dense sets, then that property is generic. Thus the phase space of the planar pendulum is two-dimensional, consisting of the position (angle) and velocity. In these cases the real space maps, in a rather abstract way, to an inverse space, which is comprised of continuous and discrete parts and evolves linearly in time. www.faqs.org /faqs/sci/nonlinear-faq

Rupert Venzke - DIMACS REU Program (Summer 2002) By a smooth manifold, we roughly mean an ( infinitely) differentiable, locally Euclidean Hausdorff space that is "second countable." In other words, the space under consideration has just about all of the properties one would normally expect of a smooth surface in R^3. Our Lie Algebra will simply be the tangent space to the Lie Group at the identity, and its representation will be the differential of ρ mapping it into the module of endomorphisms on a vector space. More generally, a Lie Algebra is essentially a vector space with a bracket product and satisfying certain important relations. dimacs.rutgers.edu /~rvenzke

Few, but ripe… » Blog Archive » Orbifolds Firstly, an n-manifold is a topological space that is Hausdorff, second countable and locally homeomorphic to In the first place, such quotient spaces will yield a technical device useful for showing the existence of hyperbolic structures on many three-manifolds. An n-dimensional orbifold is a topological space that is Hausdorff, paracompact, and locally homeomorphic to a quotient space of www.fewbutripe.com /wp/2005/06/13/orbifolds.html

Ernest Schimmerling This weakening of first countability is due to A. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. A topological space $X$ is called weakly first countable, if for every point $x$ there is a countable family $\{C_n^x \mid n\in\omega\}$ such that $x\in C_{n+1}^x \subseteq C_n^x $ and such that $U \subset X$ is open iff for each $x \in U$ some $C_n^x$ is contained in $U$. In 1976, N.N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. www.math.cmu.edu /users/eschimme/seminar/abraham1.html

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