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

Witold Marciszewski

Displaying similar documents to “A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ”

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...

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...

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.

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. ...

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.

An ordinal version of some applications of the classical interpolation theorem

Benoît Bossard (1997)

Fundamenta Mathematicae

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Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space $E_{1}$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space $E_{2}$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of $E_{1}$ and $E_{2}$ can be controlled by the Szlenk index of E, where the...

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.

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.

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...

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.

Decomposing Baire class 1 functions into continuous functions

Saharon Shelah, Juris Steprans (1994)

Fundamenta Mathematicae

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It is shown to be consistent that every function of first Baire class can be decomposed into $ℵ_{1}$ continuous functions yet the least cardinal of a dominating family in $^{ω} ω$ is $ℵ_{2}$ . The model used in the one obtained by adding $ω_{2}$ Miller reals to a model of the Continuum Hypothesis.

Filters and sequences

Sławomir Solecki (2000)

Fundamenta Mathematicae

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We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a $Π_{3}^{0}$ filter is itself $Π_{3}^{0}$ and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s...

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 Lefschetz-type coincidence theorem

Peter Saveliev (1999)

Fundamenta Mathematicae

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A Lefschetz-type coincidence theorem for two maps f,g: X → Y from an arbitrary topological space to a manifold is given: $I_{f g} = λ_{f g}$ , that is, the coincidence index is equal to the Lefschetz number. It follows that if $λ_{f g} \neq 0$ then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X,Y manifolds, f boundary-preserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with “point-like”...

More set-theory around the weak Freese–Nation property

Sakaé Fuchino, Lajos Soukup (1997)

Fundamenta Mathematicae

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We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for $ℵ_{ω}$ , we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding...

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...

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.