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# Topic: Baire category theorem

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 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. The Cantor set is a Baire space, and so is of second category in itself, but it is of first category in the interval [0, 1] with the usual topology. 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/Meagre_set   (732 words)

 Baire category theorem - Wikipedia, the free encyclopedia The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space. BCT1 is used to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle. en.wikipedia.org /wiki/Baire_category_theorem   (378 words)

 PlanetMath: Baire category theorem   (Site not responding. Last check: 2007-11-07) Alternative formulations: Call a set first category, or a meagre set, if it is a countable union of nowhere dense sets, otherwise second category. The Baire category theorem is often stated as ``no non-empty complete metric space is of first category'', or, trivially, as ``a non-empty, complete metric space is of second category''. This is version 10 of Baire category theorem, born on 2002-06-04, modified 2004-09-29. planetmath.org /encyclopedia/BaireCategoryTheorem.html   (423 words)

 Station Information - Baire space In topology and related branches of mathematics, a Baire space is a topological space in which, intuitively, there are "enough" points for certain limit processes. 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. Baire space should be contrasted with Cantor space, the set of infinite sequences of binary digits. www.stationinformation.com /encyclopedia/b/ba/baire_space.html   (532 words)

 Asian Cricket   (Site not responding. Last check: 2007-11-07) 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]. The Baire category theorem is an important tool in the study of complete spaces, such as Banach spaces and Hilbert spaces, that arise in topology and functional analysis. The proof of the Baire category theorem uses the axiom of choice; and in fact it is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice. www.asiancricket.com /index.php?title=Baire_space   (898 words)

 Uniform boundedness principle - Wikipedia, the free encyclopedia In mathematics, the uniform boundedness principle or Banach-Steinhaus Theorem is one of the fundamental results in functional analysis and, together with the Hahn-Banach theorem and the open mapping theorem, considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to boundedness. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn. en.wikipedia.org /wiki/Uniform_boundedness_principle   (290 words)

 [No title] Baire category +------------------------------------------------------------ Baire category is a measure for the size of a set in a topological space. Baire space +------------------------------------------------------------ A Baire space is a topological space with the property that the intersection of countable family of open dense subsets is dense. Baire category theorem +------------------------------------------------------------ The Baire category theorem: a complete metric space is a Baire space. www.math.harvard.edu /~knill/sofia/data/topology.txt   (1652 words)

 Encyclopedia: Baire category theorem   (Site not responding. Last check: 2007-11-07) Functional analysis In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y... In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. Theorems Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. www.nationmaster.com /encyclopedia/Baire-category-theorem   (1250 words)

 wiki/Category:Topology Definition / wiki/Category:Topology Research   (Site not responding. Last check: 2007-11-07) Banach fixed point theoremThe Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. In this theorem, n is any positive integer, and the closed unit ball is the set of all points in Euclidean n-space Rn which are at distance at most 1 from the origin.... Ham sandwich theoremThe Ham sandwich theorem, also known as the Stone-Tukey theorem in topology in mathematics, states that given n objects in n-dimensional space, it is possible to divide each one in half with a single (n − 1)-dimensional hyperplane.... www.elresearch.com /wiki/Category:Topology   (2256 words)

 PlanetMath: proof for one equivalent statement of Baire category theorem   (Site not responding. Last check: 2007-11-07) First, let's assume Baire's category theorem and prove the alternative statement. "proof for one equivalent statement of Baire category theorem" is owned by gumau. This is version 5 of proof for one equivalent statement of Baire category theorem, born on 2003-12-01, modified 2005-01-29. planetmath.org /encyclopedia/ProofOfAConsequenceOfBaireCategoryTheorem.html   (127 words)

 Baire space: Definition and Links by Encyclopedian.com - All about Baire space   (Site not responding. Last check: 2007-11-07) In topology, a Baire space is a particular type of topological space in which, intuitively, there are "enough" points for certain limit processes. Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point. 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)

 An Introduction to Banach Space Theory The Eberlein-Smulian theorem is obtained in this section, as is the result due to Krein and Smulian that the closed convex hull of a weakly compact subset of a Banach space is itself weakly compact. Schauder's theorem relating the compactness of a bounded linear operator to that of its adjoint is presented, as is the characterization of operator compactness in terms of the bounded-weak*-to-norm continuity of the adjoint. Gantmacher's theorem is obtained, as well as the equivalence of the weak compactness of a bounded linear operator to the weak*-to-weak continuity of its adjoint. www.math.lsa.umich.edu /~meggin/ibst.html   (2875 words)

 Baire Category Theorem   (Site not responding. Last check: 2007-11-07) [funct-an/9205001] A Baire Category Approach to the Bang-Bang Property... AMCA: The Baire category theorem for separately open sets. There are many ways to state the Baire Category Theorem. www.scienceoxygen.com /math/633.html   (159 words)

 Baire category theorem   (Site not responding. Last check: 2007-11-07) Atlas: The Baire category theorem for separately open sets. Minimization of Functionals of the Gradient by Baire's Theorem... Category of density points of fat Cantor sets... www.scienceoxygen.com /math/529.html   (171 words)

 Banach fixed point theorem -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-07) The theorem is named after (Click link for more info and facts about Stefan Banach) Stefan Banach (1892-1945), and was first stated by Banach in 1922. However, if the space X is (A small cosmetics case with a mirror; to be carried in a woman's purse) compact, then this weaker assumption does imply all the statements of the theorem. When using the theorem in practice, the most difficult part is typically to define X properly so that T actually maps elements from X to X, i.e. www.absoluteastronomy.com /encyclopedia/B/Ba/Banach_fixed_point_theorem.htm   (510 words)

 Nowhere dense set - Wikipedia, the free encyclopedia The union of countably many nowhere dense sets, however, need not be nowhere dense. (Thus, the nowhere dense sets need not form a sigma-ideal.) Instead, such a union is called a set of first category. The concept is important to formulate the Baire category theorem. www.wikipedia.org /wiki/Nowhere_dense   (396 words)

 Topology - Wikipedia It associates "discrete" more computable invariants to maps and spaces, often in a functorial way (see category theory). Occasionly, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories. nostalgia.wikipedia.org /wiki/Topology   (941 words)

 Functional Analysis Baire category theorem, proven both for complete metric and also locally compact Hausdorff spaces. All these constructs are treated as initial or final objects in some categories of diagrams, in order to see how this trick gives a trivial proof of uniqueness. The relevance of this to functional analysis is that projective and direct limits really do play a much larger role than often imagined, and this viewpoint can be a big help in understanding proofs. www.math.umn.edu /~garrett/m/fun   (581 words)

 Vasco Brattka's Papers   (Site not responding. Last check: 2007-11-07) We study different computable versions of Baire's Category Theorem in computable analysis. Similarly, as in constructive analysis, different logical forms of this theorem lead to different computational interpretations. We demonstrate that, analogously to the classical theorem, one of the computable versions of the theorem can be used to construct interesting counterexamples, such as a computable but nowhere differentiable function. www.informatik.fernuni-hagen.de /thi1/vasco.brattka/publications/baire.html   (79 words)

 GraduateProgram: Math, ASU   (Site not responding. Last check: 2007-11-07) Ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, field of quotients, prime and maximal ideals, characteristic, matrix rings, Euclidean rings, polynomial rings, unique factorization theorems, extension fields, degree of an extension, roots of polynomials, finite fields. Perron-Frobenius Theorem and discussion of dynamics of x(n+1)=L x)(n) for low dimensional nonnegative Leslie matrix L. Population dynamics of interacting species. Derivative, mean value theorem, Taylor's theorem; Riemann integral and integrability, fundamental theorem of calculus; exponential, logarithmic, trigonometric functions; derivative and Riemann integral of uniformly convergent sequences, power series. math.asu.edu /%7Egrad/doc/syllabi.html   (1380 words)

 Encyclopedia: Open mapping theorem   (Site not responding. Last check: 2007-11-07) In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y, then A is an open map (i.e. : Y → X is continuous as well (this is called the inverse mapping theorem). In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. www.nationmaster.com /encyclopedia/Open-mapping-theorem   (263 words)

 Baire Category Theorem In particular, we will need the theorem that says a closed subspace of a compact space is compact. This theorem fails when the collection of dense open sets is uncountable. The interval [0,1] is compact, hausdorff, a complete metric space, the nicest space you could think of, and the complement of any point is a dense open set, yet the intersection of these sets is empty. www.mathreference.com /top-ms,bct.html   (621 words)

 Standard Generals Questions   (Site not responding. Last check: 2007-11-07) Prove the structure theorem for finitely generated abelian groups (or modules over a principal ideal domain). Prove Dini's Theorem: if a sequence \$f_n\$ of functions on a closed interval is pointwise decreasing and pointwise convergent to 0, show that the convergence is uniform. Give a counterexample to Dini's theorem when the sequence is not pointwise decreasing. www.princeton.edu /%7Emissouri/Generals/generals/kiran.html   (244 words)

 6th Annual Graduate Student Conference in Logic However the category of Modal algebras and Kripke frames are not dual in the sense of category theory. Through, connections with the category of Modal Algebras, I have proved certain properties for classes of sv-frames, such as Hessney-Milner property, and also I have used the dulaity, to prove certain "definability/determinacy" conditions for classes of modal algebras. One standard formulation of the Baire Category Theorem —"The intersection of a countable number of dense open sets is dense"—is provable in RCA_0; the proof follows directly from the definition of a real number in RCA_0. www.math.uiuc.edu /~jamoreno/6thAGSCL   (1081 words)

 Graduate Mathematics Courses   (Site not responding. Last check: 2007-11-07) Sylow's theorems for finite groups, p-groups, abelian groups, group action on sets, domains, prime and maximal ideals, unique factorization domain. Limits of functions, continuity, uniform continuity, differentiation, the mean value theorem, Rolle's theorem, L'Hospital's rule, Taylor's theorem, Riemann Integral, properties of the Riemann Integral, the fundamental theorem of calculus, pointwise and uniform convergence, applications of uniform convergence. Banach spaces; the dual topology and weak topology; the Hahn-Banach, Krein- Milman and Alaoglu theorems; the Baire category theorem; the closed graph theorem; the open mapping theorem; the Banach-Steinhaus theorem; elementary spectral theory; and differential equations. donaldson.math.howard.edu /%7Ereb/gradcour.htm   (1023 words)

 Office of the Provost and Chief Academic Officer Freihet space of holomorphic functions, Montel’s theorem, normal families, Picard’s theorem, Mittag-Leffler’s theorem, Weierstrass’ theorem, simply connected domains, d-bar equation and Runges Theorem, compact Riemann surfaces, de Rham Cohomology, Zeta functions, Marmonic and subharmonic functions, Dirichlet problems. Banach spaces; the dual topology and weak topology; the Hahn-Banach, Krein-Milman and Alaoglu theorems; the Baire category theorem; the closed graph theorem; the open mapping theorem; the Banach-Steinhaus theorem; elementary spectral theory; and differential equations. First-order partial differential equations, method of characteristics; Cauchy-Kovalevskaya theorem; second-order equations, classification existence, and uniqueness results; formulation of some of the classical problems of mathematical physics. www.provost.howard.edu /provost/bulletin2/g/v2gmath_b.htm   (410 words)

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