
	Where results make sense

Topic: Baire space

Related Topics

First category

Complete space

Baire category theorem

Operator norm

Irrational number

Square root

Van der Waals Bonding

Complete space

	Asian Cricket (Site not responding. Last check: 2007-11-07)
		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 property of being a Baire space is a topological property, i.e., it is preserved by homeomorphisms.
		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.asiancricket.com /index.php?title=Baire_space (898 words)

	Baire category theorem - Wikipedia, the free encyclopedia
		More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is 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.
	en.wikipedia.org /wiki/Baire_category_theorem (378 words)

	Baire space (Site not responding. Last check: 2007-11-07)
		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.sciencedaily.com /encyclopedia/baire_space (588 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.
		It is known that a T4 space with a countable basis is metrizable.
	www.math.harvard.edu /~knill/sofia/data/topology.txt (1652 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.
		That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty.
	en.wikipedia.org /wiki/Locally_compact_space (1326 words)

	PlanetMath: Baire category theorem (Site not responding. Last check: 2007-11-07)
		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''.
		In functional analysis, this important property of complete metric spaces forms the basis for the proofs of the important principles of Banach spaces: the open mapping theorem and the closed graph theorem.
		This is version 10 of Baire category theorem, born on 2002-06-04, modified 2004-09-29.
	planetmath.org /encyclopedia/BaireCategoryTheorem.html (423 words)

	Encyclopedia article on Topological space [EncycloZine] (Site not responding. Last check: 2007-11-07)
		A linear graph is a topological space that generalises many of the geometric aspects of graphs with vertices and edges.
		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 space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated.
	encyclozine.com /Topological_space (2350 words)

	F98Piotrow (Site not responding. Last check: 2007-11-07)
		Saint Raymond (1983) proved that: Separable Baire spaces are Namioka, Tychonoff Namioka spaces are Baire, and in the class of metric spaces Namioka and Baire spaces coincide.
		A space Y is co-Namioka if for any Baire space X and any metric space M and for any separately continuous function f, (*) hols.
		For example, Corson-compact spaces are co-Namioka, whereas beta-N (the Stone-Cech compactification of N) is not.
	www.math.wvu.edu /~kcies/Coll98F/F98Piotrow.html (637 words)

	Volume 24, Number 1, 1998
		It is shown that the construct of supertopological spaces and continuous maps is topological.
		A theorem proved by Fort in 1951 says that an upper or lower semi-continuous set-valued mapping from a Baire space A into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense G
		A new class of regular spaces, called CE-regular spaces, is introduced; the class of all OCE-regular spaces of J. Porter and C. Votaw [29] (and, hence, the class of all regular-closed spaces) is its proper subclass.
	www.math.bas.bg /~serdica/n1_98.html (1107 words)

	Letter B
		A nonempty metric space (PL) cannot be a,I>union (PL) of countable (PL) family of nowhere dense spaces (PL).
		A pointwise-bounded (PL) family of continuous (PL) linear operators (PL) from a Banach space (PL) to a normed space (PL) is uniformly bounded.
		The number of base vectors is the dimension of the vector space and may be infinite.
	members.fortunecity.com /jonhays/letterB.htm (1785 words)

	Homepage of Henryk Michalewski
		I proved that all separable, metrizable, zero--dimensional spaces which are of the first category in itself and which are locally coanalytic complete, are homeomorphic.
		As a corollory I proved that the space of all compact subspaces of rationals, endowed with Hausdorff metric, is a topological group.
		I gave two examples of a space with the property that the spaces $C_p(X)$ and $C_p(X\times\omega)$ are linear homeomorphic, but the spaces $C_p(X)$ and $C_p(X\times(\omega+1))$ are not linear homeomorphic, where $\omega$ and $\omega+1$ are countable ordinals equiped with ordinal topology.
	www.mimuw.edu.pl /~henrykm (659 words)

	Ondrej Kalenda - Research Papers and Publications (Site not responding. Last check: 2007-11-07)
		Assuming the consistency of the existence of a measurable cardinal, it is consistent to have two Banach spaces, X,Y, where X is a weak Asplund space such that X* (in the weak* topology) in not in Stegall's class, whereas Y* is in Stegall's class but is not weak* fragmentable.
		We prove in particular that the dual unit ball of a Banach space X is a Corson compact provided X is of the form C(K) where K is a continuous image of a Valdivia compact space, and the dual unit ball of every subspace of X is Valdivia compact.
		We prove in particular that if X is a hereditarily Baire space which has the tightness less than the least weakly inaccessible cardinal and each (closed) subspace of Xhas the countable chain condition, then every Borel class one map of X into a metric space M has the point of continuity property.
	www.karlin.mff.cuni.cz /~kalenda/publikac.htm (2291 words)

	Volume 23, Number 3-4, 1997
		Let E be an infinite dimensional separable space and for e\in E and X a nonempty compact convex subset of E, let q(e) be the metric antiprojection of e on X.
		The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2].
		If a Banach space E cannot be decomposed into a direct sum of separable and reflexive subspaces, then there exists a normed space Z and a linear continuous bijective operator T:E\to Z such that T^{-1} is not a Borel map.
	www.math.bas.bg /~serdica/n34_97.html (1162 words)

	Baire space : Second category (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.
		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.city-search.org /se/second-category.html (434 words)

	Jayne: Space of Baire functions. I
		The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either
		In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces with index
		The sequential stability index for linear subspaces of continuous real-valued functions on a compact space is shown to be invariant under isomorphic embeddings in the space of continuous real-valued functions on any compact space.
	www.numdam.org /item?id=AIF_1974__24_4_47_0 (389 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		E_0 is the equivalence relation on Baire space (functions from naturals to naturals) that says f E_0 g
		By "E_0-many" I mean the cardinality of the quotient space of Baire space by this equivalence relation.
		Given AC, of course this is the same as the cardinality of the continuum, but in models of ZF+AD it's bigger.
	www.math.niu.edu /~rusin/known-math/01_incoming/E_0 (169 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		] A measure defined on the class of all Baire sets such that the measure of any closed, compact set is finite.
		] The theorem that a complete metric space is of second category; equivalently, the intersection of any sequence of open dense sets in a complete metric space is dense.
		] A topological space in which every countable intersection of dense, open subsets is dense in the space.
	www.accessscience.com /Dictionary/B/B3/DictB3.html (3131 words)

	question.html
		If a normed space satisfies the continuous Hahn-Banach property, then the closed unit ball of its continuous dual is convex-compact in the weak* topology (this idea is in [Lu69]).
		Is there a computable mapping f such that for every finite dimensional uniformly smooth normed space E, with modulus of smoothness r, the map f(r) is a witness of the fact that E does not satisfy the finite tree property.
		Is there a computable mapping f such that for every finite dimensional normed space E, if t is a witness of the fact that E does not satisfy the finite tree property, then f(t) is a witness of the fact that the continuous dual E' does not satisfy the finite tree property.
	www2.univ-reunion.fr /~mar/question.html (1466 words)

	Nonlinear Science FAQ
		Thus the phase space of the planar pendulum is two-dimensional, consisting of the position (angle) and velocity.
		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.
		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 (11547 words)

	PlanetMath: Baire space (Site not responding. Last check: 2007-11-07)
		A Baire space is a topological space such that the intersection of any countable family of open and dense sets is dense.
		This is version 2 of Baire space, born on 2003-06-11, modified 2004-02-27.
		(General topology :: Spaces with richer structures :: Baire category, Baire spaces)
	planetmath.org /encyclopedia/BaireSpace.html (54 words)

	Ondrej Kalenda (Site not responding. Last check: 2007-11-07)
		Let X be a topological space, M a metric space.
		A classical result of Baire states that if X and M are Polish spaces, then every function of the first Borel class of X to M has the point of continuity property.
		If there is a measurable cardinal, then there is a hereditarily t-Baire space X and an \left(\Cal F\wedge\Cal G\right) -measurable function on X which has no point of continuity.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/49.htm (156 words)

	Array convergence of functions of the first Baire-class
		This result generalizes easily to Hausdorff spaces which satisfy the first countability axiom.
		In angelic spaces the notions of (relative) compactness, (relative) countable compactness and (relative) sequential compactness coincide.
		A device of R. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc.
	www.math.utep.edu /Faculty/helmut/baire/baire.html (753 words)

	CONTENTS - Vol (Site not responding. Last check: 2007-11-07)
		We prove that the space N(E) of all norms on n-dimensional linear space E is a proper G-space under the natural action of the full linear group G = GL(n).The Banach-Mazur compactum Q (n) is just the orbit space of the G- space N (E).
		We generalize this fact by showing that for a locally compact almost connected group G, its maximal compact subgroup K, and a proper G-space X, X/G is an A (N) R space provided it is metrizable and X is a K-A (N) R space.
		We extend the theory of semifractals to arbitrary metric spaces.
	www.pan.pl /bulletin/MATH/m2-98.htm (508 words)

	Edwin Hewitt as Topologist: An Appreciation by W. W. Comfort
		There are easily stated specific questions related to the very general problem "Which spaces are resolvable?" Hewitt's attention to spaces which are not necessarily completely regular or even Hausdorff legitimatized the subsequent study of resolvability in "peculiar" spaces of various sorts.
		Obviously X cannot be a Tychonoff space; Urysohn [70] and Pospísil [60] found Hausdorff spaces X, countably infinite and of arbitrary pre-assigned infinite cardinality respectively, of this type, but their examples are not regular.
		509] that a space with the properties of Y cannot in addition be regular, since as noted by Urysohn [70] every connected regular (Hausdorff) space is uncountable.
	at.yorku.ca /t/o/p/d/08.htm (5592 words)

	No Title (Site not responding. Last check: 2007-11-07)
		Cardinality, Lebesgue measure and Baire categories are the most common.
		In measure spaces, sets with null measure are usually called negligible and in a Baire space, first category sets can be thought of as small sets.
		We shall review some generic results in Analysis (continuous functions are nowhere differentiable, uniqueness in Peano's problem, existence of fixed points for nonexpansive mappings, etc) and show an easy example about the generic size of the set of zeros for a continuous function where all the above notions of size are involved.
	www.math.utep.edu /events/coll-abstracts/040999 (155 words)

	Kids.net.au - Encyclopedia Category - (Site not responding. Last check: 2007-11-07)
		Kant uses the word for the preconceived notions we use to organize sense data
		a collection of mathematical objects of the same kind, together with the structure-preserving maps between them; as in "the category of all topological spaces and continuous maps".
		in topology, one distinguishes between sets of first and second category; see Baire space.
	www.kids.net.au /encyclopedia-wiki/ca/Category (170 words)

Try your search on: Qwika (all wikis)