An ordinal version of some applications of the classical interpolation theorem

Benoît Bossard

Displaying similar documents to “An ordinal version of some applications of the classical interpolation theorem”

Dugundji extenders and retracts on generalized ordered spaces

Gary Gruenhage, Yasunao Hattori, Haruto Ohta (1998)

Fundamenta Mathematicae

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For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{c h}$ -extender (resp. $L_{c c h}$ -extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous $L_{c h}$ -extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology...

Operators on C(ω^α) which do not preserve C(ω^α)

Dale Alspach (1997)

Fundamenta Mathematicae

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It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C (ω^{ω^{α}})$ onto itself such that if Y is a subspace of $C (ω^{ω^{α}})$ which is isomorphic to $C (ω^{ω^{α}})$ , then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C (ω^{ω^{α}})$ onto itself there is a subspace of $C (ω^{ω^{α}})$ which is isomorphic to $C (ω^{ω^{α}})$ on which the operator is an isomorphism.

Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Katsuro Sakai, Shigenori Uehara (1999)

Fundamenta Mathematicae

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Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by $U S C C_{B} (X)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $φ \in U S C C_{B} (X)$ with its graph which is a closed subset of X × ℝ. The space $U S C C_{B} (X)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $U S C C_{B} (X)$ is homeomorphic...

A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Witold Marciszewski (1997)

Fundamenta Mathematicae

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We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions $c_{p} (X)$ onto $c_{p} (X)$ × ℝ $. I n p a r t i c u l a r,$ cp(X) $i s n o t l i n e a r l y h o m e o m o r p h i c t o$ cp(X) $\times ℝ$ . One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

Homotopy orbits of free loop spaces

Marcel Bökstedt, Iver Ottosen (1999)

Fundamenta Mathematicae

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Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors $Ω_{λ} (R)$ and ℓ(R) together with algebra homomorphisms $e : Ω_{λ} (R) \to H * (Λ X)$ and $ψ : ℓ (R) \to H * (E S^{1} \times_{S^{1}} Λ X)$ . When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

The Zahorski theorem is valid in Gevrey classes

Jean Schmets, Manuel Valdivia (1996)

Fundamenta Mathematicae

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Let Ω,F,G be a partition of $ℝ^{n}$ such that Ω is open, F is $F_{σ}$ and of the first category, and G is $G_{δ}$ . We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.

The space of ANR’s in $ℝ^{n}$

Tadeusz Dobrowolski, Leonard Rubin (1994)

Fundamenta Mathematicae

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The hyperspaces $A N R (ℝ^{n})$ and $A R (ℝ^{n})$ in $2^{ℝ^{n}} (n \geq 3)$ consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute $G_{δ σ δ}$ -spaces and that, indeed, they are not $F_{σ δ σ}$ -spaces. The main result is that $A N R (ℝ^{n})$ is an absorber for the class of all absolute $G_{δ σ δ}$ -spaces and is therefore homeomorphic to the standard model space $Ω_{3}$ of this class.

For almost every tent map, the turning point is typical

Henk Bruin (1998)

Fundamenta Mathematicae

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Let $T_{a}$ be the tent map with slope a. Let c be its turning point, and $μ_{a}$ the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, $ʃ g d μ_{a} = l i m_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} g (T_{a}^{i} (c))$ . As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

Universal spaces in the theory of transfinite dimension, II

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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We construct a family of spaces with “nice” structure which is universal in the class of all compact metrizable spaces of large transfinite dimension $ω_{0}$ , or, equivalently, of small transfinite dimension $ω_{0}$ ; that is, the family consists of compact metrizable spaces whose transfinite dimension is $ω_{0}$ , and every compact metrizable space with transfinite dimension $ω_{0}$ is embeddable in a space of the family. We show that the least possible cardinality of such a universal family is equal to the...

Cantor manifolds in the theory of transfinite dimension

Wojciech Olszewski (1994)

Fundamenta Mathematicae

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For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space $Z_{α}$ such that $i n d Z_{α} = α$ , and no closed subset L of $Z_{α}$ with ind L less than the predecessor of α is a partition in $Z_{α}$ . An α-dimensional Cantor Ind-manifold can be constructed similarly.

The minimum uniform compactification of a metric space

R. Grant Woods (1995)

Fundamenta Mathematicae

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It is shown that associated with each metric space (X,d) there is a compactification $u_{d} X$ of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of $u_{d} X$ are presented, and a detailed study of the structure of $u_{d} X$ is undertaken. This culminates in a topological characterization of the outgrowth $u_{d} ℝ^{n} ∖ ℝ^{n}$ , where $(ℝ^{n}, d)$ is Euclidean n-space with its usual metric. ...

Property C'', strong measure zero sets and subsets of the plane

Janusz Pawlikowski (1997)

Fundamenta Mathematicae

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Let X be a set of reals. We show that • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections $F_{x}$ (x ∈ X) null, $\cup_{x \in X} F_{x}$ is null; • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections $F_{x}$ (x ∈ ℝ) null, $\cup_{x \in X} F_{x}$ is null.

On compact spaces carrying Radon measures of uncountable Maharam type

David Fremlin (1997)

Fundamenta Mathematicae

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If Martin’s Axiom is true and the continuum hypothesis is false, and X is a compact Radon measure space with a non-separable $L^{1}$ space, then there is a continuous surjection from X onto ${[0, 1]}^{ω_{1}}$ .

A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

W. Jurkat, D. Nonnenmacher (1994)

Fundamenta Mathematicae

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Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^{n}$ , we are led to an integration process over quite general sets $A \subseteq q ℝ^{n}$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

Countable partitions of the sets of points and lines

James Schmerl (1999)

Fundamenta Mathematicae

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The following theorem is proved, answering a question raised by Davies in 1963. If $L_{0} \cup L_{1} \cup L_{2} \cup . . .$ is a partition of the set of lines of $ℝ^{n}$ , then there is a partition $ℝ^{n} = S_{0} \cup S_{1} \cup S_{2} \cup . . .$ such that $| ℓ \cap S_{i} | \leq 2$ whenever $ℓ \in L_{i}$ . There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.

Construction of non-subadditive measures and discretization of Borel measures

Johan Aarnes (1995)

Fundamenta Mathematicae

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The main result of the paper provides a method for construction of regular non-subadditive measures in compact Hausdorff spaces. This result is followed by several examples. In the last section it is shown that “discretization” of ordinary measures is possible in the following sense. Given a positive regular Borel measure λ, one may construct a sequence of non-subadditive measures $μ_{n}$ , each of which only takes a finite set of values, and such that $μ_{n}$ converges to λ in the w*-topology. ...

Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

Alessandro Andretta, Alberto Marcone (1997)

Fundamenta Mathematicae

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We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $\prod_{2}^{0}$ -complete and that the set of Cauchy problems which locally have a unique solution is $\sum_{3}^{0}$ -complete. We prove that the set of Cauchy problems which have a global solution is...

A partial order where all monotone maps are definable

Martin Goldstern, Saharon Shelah (1997)

Fundamenta Mathematicae

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It is consistent that there is a partial order (P,≤) of size $ℵ_{1}$ such that every monotone function f:P → P is first order definable in (P,≤).

From Newton’s method to exotic basins Part I: The parameter space

Krzysztof Barański (1998)

Fundamenta Mathematicae

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This is the first part of the work studying the family $𝔉$ of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of $𝔉$ and give a detailed study of the subfamily $ℱ_{2}$ consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in $ℱ_{2}$ from Newton maps to maps with so-called exotic basins.