Choquet simplexes whose set of extreme points is $K$-analytic

Michel Talagrand

Displaying similar documents to “Choquet simplexes whose set of extreme points is $K$ -analytic”

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

On the complexity of subspaces of $S_{ω}$

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

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Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^{X}$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{ω}$ is $F_{σ δ}$ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{ω}$ . We show that $S_{ω}$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....

The Lindelöf property and σ-fragmentability

B. Cascales, I. Namioka (2003)

Fundamenta Mathematicae

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In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space ${[- 1, 1]}^{D}$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of ${[- 1, 1]}^{D}$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is...

Applications of some results of infinite-dimensional topology to the topological classification of operator images

Taras Banakh, Tadeusz Dobrowolski, Anatoliĭ Plichko

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This volume consists of three relatively independent articles devoted to the topological study of the so-called operator images and weak unit balls of Banach spaces. These articles are: “The topological classification of weak unit balls of Banach spaces” by T. Banakh, “The topological and Borel classification of operator images” by T. Banakh, T. Dobrowolski and A. Plichko, and “Operator images homeomorphic to $Σ^{ω}$ ” by T. Banakh. The articles summarize investigations that has been done by...

The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ( $R^{r}$ )

V. Farmaki (1987)

Fundamenta Mathematicae

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Measure preserving analytic diffeomorphisms of countable dense sets in $C^{n}$ and $R^{n}$

Michał Morayne (1987)

Colloquium Mathematicae

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Banach spaces of bounded Szlenk index

E. Odell, Th. Schlumprecht, A. Zsák (2007)

Studia Mathematica

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For a countable ordinal α we denote by $_{α}$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each $_{α}$ admits a separable, reflexive universal space. We also show that spaces in the class $_{ω^{α \cdot ω}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each $_{α}$ is analytic in the Effros-Borel structure of subspaces of C[0,1].

Pointwise convergence and the Wadge hierarchy

Alessandro Andretta, Alberto Marcone (2001)

Commentationes Mathematicae Universitatis Carolinae

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We show that if $X$ is a $Σ_{1}^{1}$ separable metrizable space which is not $σ$ -compact then $C_{p}^{*} (X)$ , the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{1}^{1}$ -complete. Assuming projective determinacy we show that if $X$ is projective not $σ$ -compact and $n$ is least such that $X$ is $Σ_{n}^{1}$ then $C_{p} (X)$ , the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel- $Π_{n}^{1}$ -complete. We also prove a simultaneous improvement of theorems...

Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara (2015)

Fundamenta Mathematicae

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Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$ . Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $κ^{κ}$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in $κ^{κ}$ for uncountable regular κ is however consistent (with...

On the structure of non-dentable subsets of $C (ω^{ω^{k}})$

Pericles D. Pavlakos, Minos Petrakis (2011)

Studia Mathematica

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It is shown that there is no closed convex bounded non-dentable subset K of $C (ω^{ω^{k}})$ such that on subsets of K the PCP and the RNP are equivalent properties. Then applying the Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains a non-dentable subset L so that on L the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of $C (ω^{ω^{k}})$ .

Displaying similar documents to “Choquet simplexes whose set of extreme points is K -analytic”

Displaying similar documents to “Choquet simplexes whose set of extreme points is $K$ -analytic”