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Topic: Baire property

	PlanetMath: Baire category theorem
		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 (416 words)

	Springer Online Reference Works
		Families of real functions which are defined inductively using the ordinal number of limit transitions involved in the definition of the function, and which constitute the classification of functions proposed in 1899 by R.
		He showed that a necessary and sufficient condition for a discontinuous function to belong to the first class is the existence of a point of continuity of the induced function on each perfect set (Baire's theorem).
		A modern English reference for the notion of Baire classes is [a1].
	eom.springer.de /b/b015030.htm (469 words)

	Publications of Rafal Filipow ::: Details
		We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property.
		That is, every function for which f(x+h)-f(x) has the Baire property for every real h is of the form f=g+A where g has the Baire property and A is additive.
		It is consistent with ZFC that the family of functions with the Baire property has the difference property.
	math.univ.gda.pl /~rfilipow/papers/details.html (549 words)

	Springer Online Reference Works
		The class of sets with the Baire property is closed under the operations of complementing, taking countable unions and taking countable intersections.
		For an example of a set which does not have the Baire property, see [1].
		A set with the Baire property is often called a Baire set or an almost-open set.
	eom.springer.de /B/b015040.htm (156 words)

	Lockport Union-Sun & Journal Online - Few turn out to challenge city assessments (Site not responding. Last check: 2007-10-15)
		A handful of property owners who were first offered an assessment cut in 2006, then had it canceled, turned out to have their say Tuesday at City Hall.
		Several property owners grieved assessments on the basis that the prices they paid in auctions, foreclosure and estate sales are closer to their true worth.
		Baire, who acted as consultant to Acting Assessor Dick Mullaney during the proceedings, checked property data and found that same house had sold for almost $74,000 in October 2003.
	www.lockportjournal.com /local/local_story_172002221.html (911 words)

	Antimeta: October 2006 Archives
		Intuitively, there is a sense in which a dense open subset of a set is "almost equal" to that set (this is in Baire's sense, rather than the senses of Borel or Lebesgue), and there is also a sense in which open sets are extremely well-behaved.
		Baire's idea is that a dense open subset of a nice topological space is almost all of it.
		And as a further blow against the Baire notion, the central result of Dougherty and Foreman is a Banach-Tarski-style decomposition where every set involved has the property of Baire.
	www.ocf.berkeley.edu /~easwaran/blog/2006/10 (1577 words)

	The barrelled propert of function spaces $C_p(Y\|X)$ and $C_k(Y\|X)$ by Hui Teng, Shou Lin and Chuan Liu
		For example, it was used by Lutzer and McCoy in [5] to characterize the Baire property of C
		The barrelled property is an important topic of functional analysis.
		(Y) respectively, the Baire property and the barrelled property of the former imply that of the latter.
	at.yorku.ca /b/a/a/e/13.htm (251 words)

	Property of Baire - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-15)
		A subset A of a topological space X has the property of Baire (Baire property) if it differs from an open set by a meager set; that is, if there is an open
		If a subset of a Polish space has the property of Baire, then its corresponding Banach-Mazur game is determined.
		It follows from the axiom of choice that there are sets of reals without the property of Baire.
	en.wikipedia.org.cob-web.org:8888 /wiki/Property_of_Baire (187 words)

	Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter
		The Lebesgue-measurable sets correspond to the sets with the Baire property -- i.e., the sets which are the symmetric difference of an open set and a meager set.
		A couple of very weak consequences of the Axiom of Choice are the existence of (i) subsets of R which are not Lebesgue measurable, and (ii) subsets of R which lack the Baire property.
		Let BP be the statement that ``every subset of R has the Baire property''; this is a very strong negation of the Axiom of Choice.
	at.yorku.ca /z/a/a/b/18.htm (848 words)

	Amazon.com: "Property of Baire": Key Phrase page (Site not responding. Last check: 2007-10-15)
		A set is said to have the Property of Baire if it differs from a Borel set by a meager set; that is, the symmetric difference X o B (=...
		Their main result was that the Banach-Tarski paradox could be performed using pieces with the property of Baire [Dougherty and Foreman 94].
		An example of such a measure is the product measure in the Cantor space {0,1}W. The Property of Baire In Chapter 4 we proved the Baire Category Theorem (Theorem 4.8): The intersection of countably many dense open sets of...
	www.amazon.com /phrase/Property-of-Baire (516 words)

	arxivmath's Journal
		There is a well-known global equivalence between \Sigma^1_2 sets having the Universal Baire property, two-step \Sigma^1_3 generic absoluteness, and the closure of the universe under the sharp operation.
		In their paper, the authors raise the question whether a geodesic metric space with the property that the intersection of any two closed balls has eccentricity 0, is necessarily a real tree.
		In this article, we study global analytic properties of the Poincare mapping, in particular, its analytic continuation, its singularities and its fixed points.
	arxivmath.livejournal.com /2006/03/23 (2060 words)

	Citations: Can you take Solovay's inaccessible away - Shelah (ResearchIndex)
		The oe algebra of sets with the property of Baire is very large.
		....there was a model of ZFC in which projective sets of reals were measurable and had Baire property (cf [So] Solovay s model was built under the assumption that there was an inaccessible cardinal.
		The property is preserved under the amalgamation and the iteration with Amoeba Algebra for Category.
	citeseer.ist.psu.edu /context/217937/0 (730 words)

	[No title]
		A corollary of this is that any Hamel basis with the Baire property is a first category set.
		(The Baire category analog of this result is given in #2 above.) The Baire category version of this result [There exists a Hamel basis such that neither it nor its complement contains a second category set.], is given in Marek Kuczma, "On some properties of Erdos sets", Colloq.
		So the basis is totally imperfect, and thus non-measurable and without the property of Baire.
	www.math.niu.edu /~rusin/known-math/99/hamel (3445 words)

	Homepage of Henryk Michalewski
		Game--theoretic approach to the hereditary Baire property of $C_p({\mathbb N}_F)$, Bull.
		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.
		This construction appeals to the affinity between the o-bounded property and Menger property of a given topological group.
	www.mimuw.edu.pl /~henrykm (717 words)

	MATHnetBASE: Mathematics Online (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-15)
		These include functions without the Baire property, functions associated with a Hamel basis of the real line, and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum.
		While the theory of such equations is well established, the study of their stability properties has grown rapidly only in the past 20 years, and most results have remained scattered in journals and conference proceedings.
		This includes their general properties, conditions that allow points of n-fold symmetric products to be arcwise accessible from their complement, points that arcwise disconnect the n-fold hyperspaces, the n-fold hyperspaces of graphs, and theorems relating n-fold hyperspaces and cones.
	www.mathnetbase.com.cob-web.org:8888 /ejournals/info/whatnew.asp (11243 words)

	Ondrej Kalenda - Research Papers and Publications
		We investigate Baire-one functions whose graph is contained in a graph of usco mapping.
		Namely, it yields an example of an affine continuous image of a convex Valdivia compact (in the weak* topology of a dual Banach space) which is not Valdivia, and shows that the property of the dual unit ball being Valdivia is not an isomorphic property.
		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 (2698 words)

	Filtrations of random processes in the light of classification theory. I (Site not responding. Last check: 2007-10-15)
		Filtered probability spaces (called "filtrations" for short) are shown to satisfy such a topological zero-one law: for every property of filtrations, either the property holds for almost all filtrations, or its negation does.
		The set of all isomorphic classes of filtrations may be identified with the orbit space X/G for a special Polish G-space X.
		A "property of filtrations" means a G-invariant subset of X having the Baire property.
	www.tau.ac.il /~tsirel/Research/Recent/classif.html (204 words)

	On Shelah's amalgamation (ResearchIndex) (Site not responding. Last check: 2007-10-15)
		Abstract: The aim of this paper is to present a detailed explanation of three models of Shelah.
		We show the rule of the amalgamation in the construction of models in which all definable sets of reals have Baire property or are Lebesgue measurable.
		Next we construct a model in which every projective set of reals has Baire property, a model with the Uniformization Property and a model of ZF + DC in which all subsets of the real line are Lebesgue measurable but there is a set without Baire property.
	citeseer.ist.psu.edu /judah97shelahs.html (344 words)

	Dept. Math/Stats: 1996 Calendar of Events
		The Baire property is examined in the case of finite products of a space and its dense subspace.
		In this thesis a general Riesz decomposition theorem is proved giving a unique representation for every positive charge as a sum of two elements, one belonging to a fixed set M of positive charges and the other one to its orthogonal complement.
		The asymptotic properties of constrained and unconstrained least-squares estimate of the parameters of a random amplitude sinusoid are analyzed.
	www.math.yorku.ca /CalendarOfEvents/1996calendar.html (8953 words)

	Amazon.com: "Baire Category": Key Phrase page (Site not responding. Last check: 2007-10-15)
		See all pages with references to Baire Category.
		Baire Category 8.A Meager Sets Let X be a topological space.
		A 1964 colloquium talk by Johannes de Groot from Amsterdam on cotopologies and the Baire category theorem inspired George to work on some aspects of cotopologies.
	www.amazon.com /phrase/Baire-Category (508 words)

	[No title]
		The properties of the spaces EMBED Equation.3 and EMBED Equation.3 are usually very different.
		For example, EMBED Equation.3 is a complete metric space, and therefore, has the Baire property.
		In particular, EMBED Equation.3 is neither metrizable, nor has the Baire property.
	www.users.muohio.edu /randrib/Arhangelskii.doc (518 words)

	David Gauld's home page (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-15)
		My research interests are in set theoretic topology, especially applications to non-metrisable manifolds, and topological properties of manifolds near the limit of metrisability.
		If you are interested in my collection of topological properties equivalent to metrisability for a manifold click here: metrisability.
		'Spaces with property pp' (with Paul Gartside and Abdul Mohamad), Topology and its Applications, 153/15(September 2006), 3029-3037.
	www.math.auckland.ac.nz.cob-web.org:8888 /~gauld (2607 words)

	Atlas: The Steinhaus property in topological groups by Hans Weber (Site not responding. Last check: 2007-10-15)
		for every measurable subset M of E of positive measure the intersection of H and M is non-measurable and for every non-empty open subset O of E the intersection of H and O doesn’t have the Baire property).
		Therefore the cosets of H form a decomposition of E in m “pathological” subsets.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqq-32.
	atlas-conferences.com /c/a/q/q/32.htm (329 words)

	On the absolute Baire property., John C. Morgan
		On the absolute Baire property., John C. Morgan
		[1] R. Baire, Sur les functionsde variables reelles, Annali di Matematica, Serie 3, 3 (1899), 1-123.
		[14] J. Morgan, The absolute Baire property, Pacific J. Math., 65 (1976), 421-436.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.pjm/1102806140 (498 words)

	IngentaConnect On the measurability and the Baire property of t-Wright-convex fu...
		IngentaConnect On the measurability and the Baire property of t-Wright-convex fu...
		On the measurability and the Baire property of t-Wright-convex functions
		In this note we will show that every Lebesgue measurable or Baire measurable t-Wright-convex function $$ f:(a, b) \to \mathbb{R} $$ is continuous.
	www.ingentaconnect.com /content/klu/10/2004/00000068/F0020001/art00004 (153 words)

	Atlas: Ultrafilters with property (s) by Arnold W. Miller (Site not responding. Last check: 2007-10-15)
		, then U cannot be Lebesgue measurable or have the property of Baire.
		It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajz-51.
	atlas-conferences.com /cgi-bin/abstract/cajz-51 (201 words)

	[FOM] Hahn Banach and the Baire Property (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-15)
		Previous message: [FOM] Hahn Banach and the Baire Property
		The existence of a finitely additive probability measure on P(N) which vanishes on finite sets is sufficient to give a subset of {0, 1}^N which lacks the baire property.
		Specifically (identifying {0, 1}^N with P(N) in the obvious way) the set { a \in {0, 1}^N : u(a) = 0 } lacks the baire property.
	www.cs.nyu.edu.cob-web.org:8888 /pipermail/fom/2005-March/008824.html (293 words)

	Proceedings of the American Mathematical Society
		Some consequences of these extension properties are also studied.
		A. Kamburelis, On cardinal numbers related to Baire property, preprint, Wroc
		Keywords: Measure and category, Borel sets, Baire Property, $\sigma$-algebra, $\sigma$-ideal
	www.mathaware.org /proc/2001-129-01/S0002-9939-00-05505-2/home.html (328 words)

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