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Displaying similar documents to “Normal P-spaces and the G δ -topology”

On the k -Baire property

Alessandro Fedeli (1993)

Commentationes Mathematicae Universitatis Carolinae

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In this note we show the following theorem: “Let X be an almost k -discrete space, where k is a regular cardinal. Then X is k + -Baire iff it is a k -Baire space and every point- k open cover 𝒰 of X such that card ( 𝒰 ) k is locally- k at a dense set of points.” For k = 0 we obtain a well-known characterization of Baire spaces. The case k = 1 is also discussed.

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a G δ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β ( f ) ω ξ · ω ξ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family f α : X Y ( α < ω ) of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set α < ω : f α ( x ) f ( x ) is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of Σ η sets which can be interesting in its own right.

Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík, Jiří Spurný (2012)

Studia Mathematica

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If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball B E * that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of B E * , generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire...

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense G δ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true F σ δ sets.

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space κ κ for an uncountable κ with κ < κ = κ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

A poset of topologies on the set of real numbers

Vitalij A. Chatyrko, Yasunao Hattori (2013)

Commentationes Mathematicae Universitatis Carolinae

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On the set of real numbers we consider a poset 𝒫 τ ( ) (by inclusion) of topologies τ ( A ) , where A , such that A 1 A 2 iff τ ( A 1 ) τ ( A 2 ) . The poset has the minimal element τ ( ) , the Euclidean topology, and the maximal element τ ( ) , the Sorgenfrey topology. We are interested when two topologies τ 1 and τ 2 (especially, for τ 2 = τ ( ) ) from the poset define homeomorphic spaces ( , τ 1 ) and ( , τ 2 ) . In particular, we prove that for a closed subset A of the space ( , τ ( A ) ) is homeomorphic to the Sorgenfrey line ( , τ ( ) ) iff A is countable. We study also common...

Non-separable Banach spaces with non-meager Hamel basis

Taras Banakh, Mirna Džamonja, Lorenz Halbeisen (2008)

Studia Mathematica

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We show that an infinite-dimensional complete linear space X has: ∙ a dense hereditarily Baire Hamel basis if |X| ≤ ⁺; ∙ a dense non-meager Hamel basis if | X | = κ ω = 2 κ for some cardinal κ.

Uniqueness of measure extensions in Banach spaces

J. Rodríguez, G. Vera (2006)

Studia Mathematica

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Let X be a Banach space, B B X * a norming set and (X,B) the topology on X of pointwise convergence on B. We study the following question: given two (non-negative, countably additive and finite) measures μ₁ and μ₂ on Baire(X,w) which coincide on Baire(X,(X,B)), does it follow that μ₁ = μ₂? It turns out that this is not true in general, although the answer is affirmative provided that both μ₁ and μ₂ are convexly τ-additive (e.g. when X has the Pettis Integral Property). For a Banach space Y...

Insertion of a Contra-Baire- 1 (Baire- . 5 ) Function

Majid Mirmiran (2019)

Communications in Mathematics

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Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- . 5 function between two comparable real-valued functions on the topological spaces that F σ -kernel of sets are F σ -sets.

Typical multifractal box dimensions of measures

L. Olsen (2011)

Fundamenta Mathematicae

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We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on d . We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

A model-theoretic Baire category theorem for simple theories and its applications

Ziv Shami (2013)

Fundamenta Mathematicae

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We prove a model-theoretic Baire category theorem for τ ̃ l o w f -sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in countable nfcp theories: either every type that is internal in a minimal type is essentially 1-based by means of the forking topologies, or T interprets an infinite definable 1-based group of finite D-rank or T interprets a strongly minimal formula.