Gδ -sets in topological spaces and games

Winfried Just; Marion Scheepers; Juris Steprans; Paul Szeptycki

Displaying similar documents to “Gδ -sets in topological spaces and games”

Convexity ranks in higher dimensions

Menachem Kojman (2000)

Fundamenta Mathematicae

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A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a...

On partitions of lines and space

Paul Erdös, Steve Jackson, R. Mauldin (1994)

Fundamenta Mathematicae

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We consider a set, L, of lines in $ℝ^{n}$ and a partition of L into some number of sets: $L = L_{1} \cup . . . \cup L_{p}$ . We seek a corresponding partition $ℝ^{n} = S_{1} \cup . . . \cup S_{p}$ such that each line l in $L_{i}$ meets the set $S_{i}$ in a set whose cardinality has some fixed bound, $ω_{τ}$ . We determine equivalences between the bounds on the size of the continuum, $2^{ω} \leq ω_{θ}$ , and some relationships between p, $ω_{τ}$ and $ω_{θ}$ .

Rigid $ℵ_{ε}$ -saturated models of superstable theories

Ziv Shami, Saharon Shelah (1999)

Fundamenta Mathematicae

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In a countable superstable NDOP theory, the existence of a rigid $ﬡ_{ε}$ -saturated model implies the existence of $2^{λ}$ rigid $ﬡ_{ε}$ -saturated models of power λ for every $λ > 2^{ﬡ_{0}}$ .

Strong Fubini properties of ideals

Ireneusz Recław, Piotr Zakrzewski (1999)

Fundamenta Mathematicae

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Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections $D_{x} = y : ⟨ x, y ⟩ \in D$ are in J, then the sections $D^{y} = x : ⟨ x, y ⟩ \in D$ are in I for every y outside a set from J (“measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this...

Cofinal $Σ_{1}^{1}$ and $Π_{1}^{1}$ subsets of $ω^{ω}$

Gabriel Debs, Jean Saint Raymond (1999)

Fundamenta Mathematicae

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We study properties of $\sum_{1}^{1}$ and $π_{1}^{1}$ subsets of $ω^{ω}$ that are cofinal relative to the orders ≤ (≤*) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement " $\forall α \in ω^{ω}, ω^{ω} \cap L (α)$ is bounded for ≤*".

If it looks and smells like the reals...

Franklin Tall (2000)

Fundamenta Mathematicae

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Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_{M}$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if $X_{M}$ is homeomorphic to ℝ, then $X = X_{M}$ . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.

Difference functions of periodic measurable functions

Tamás Keleti (1998)

Fundamenta Mathematicae

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We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions $Δ_{h} f (x) = f (x + h) - f (x)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ (ℱ, G) = H \subset ℝ / ℤ : (\exists f \in ℱ G) (\forall h \in H) Δ_{h} f \in G$ , we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $𝕋 = ℝ / ℤ$ that are invariant for changes on null-sets...

Locally constant functions

Joan Hart, Kenneth Kunen (1996)

Fundamenta Mathematicae

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Let X be a compact Hausdorff space and M a metric space. $E_{0} (X, M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_{0} (X, M)$ is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_{0} (X, M)$ as a normed linear space. We also build three first countable...

Misiurewicz maps unfold generically (even if they are critically non-finite)

Sebastian van Strien (2000)

Fundamenta Mathematicae

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We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if $f_{λ_{0}}$ is critically finite with non-degenerate critical point $c_{1} (λ_{0}), . . ., c_{n} (λ_{0})$ such that $f_{λ_{0}}^{k_{i}} (c_{i} (λ_{0})) = p_{i} (λ_{0})$ are hyperbolic periodic points for i = 1,...,n, then IV-1. Age impartible......................................................................................................................................................................... 31 $λ \mapsto (f_{λ}^{k_{1}} (c_{1} (λ)) - p_{1} (λ), . . ., f_{λ}^{k_{d - 2}} (c_{d - 2} (λ)) - p_{d - 2} (λ))$ is a local diffeomorphism...

Ergodicity for piecewise smooth cocycles over toral rotations

Anzelm Iwanik (1998)

Fundamenta Mathematicae

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Let α be an ergodic rotation of the d-torus $𝕋^{d} = ℝ^{d} / ℤ^{d}$ . For any piecewise smooth function $f : 𝕋^{d} \to ℝ$ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on $L^{2} (𝕋^{d})$ is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product $S_{f} : 𝕋^{d + 1} \to 𝕋^{d + 1}$ must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum...

Types on stable Banach spaces

José Iovino (1998)

Fundamenta Mathematicae

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We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever $\hat{X}$ is an ultrapower of X and B is a ball in $\hat{X}$ , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B. The preceding...

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ Kolyada, Michał Misiurewicz, L’ubomír Snoha (1999)

Fundamenta Mathematicae

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The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ${(X_{i})}_{i = 1}^{\infty}$ and a sequence of continuous maps ${(f_{i})}_{i = 1}^{\infty}$ , $f_{i} : X_{i} \to X_{i + 1}$ , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of $f_{n} ○ . . . ○ f_{2} ○ f_{1}$ . As an application we construct a large class of smooth triangular maps of the square of type $2^{\infty}$ and...

Raising dimension under all projections

John Cobb (1994)

Fundamenta Mathematicae

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As a special case of the general question - “What information can be obtained about the dimension of a subset of $ℝ^{n}$ by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in $ℝ^{3}$ each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in $ℝ^{n}$ whose images contain open sets, expanding on a result of Borsuk.

A functional S-dual in a strong shape category

Friedrich Bauer (1997)

Fundamenta Mathematicae

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In the S-category $P$ (with compact-open strong shape mappings, cf. §1, instead of continuous mappings, and arbitrary finite-dimensional separable metrizable spaces instead of finite polyhedra) there exists according to [1], [2] an S-duality. The S-dual $D X, X = (X, n) \in P$ , turns out to be of the same weak homotopy type as an appropriately defined functional dual $\bar{{(S^{0})}^{X}}$ (Corollary 4.9). Sometimes the functional object $\bar{X^{Y}}$ is of the same weak homotopy type as the “real” function space $X^{Y}$ (§5).

How to recognize a true Σ^0_3 set

Etienne Matheron (1998)

Fundamenta Mathematicae

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Let X be a Polish space, and let ${(A_{p})}_{p \in ω}$ be a sequence of $G_{δ}$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether $\cup_{p \in ω} A_{p}$ is a true $\sum_{3}^{0}$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true $\sum_{3}^{0}$ .

Nonseparable Radon measures and small compact spaces

Grzegorz Plebanek (1997)

Fundamenta Mathematicae

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We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube ${[0, 1]}^{κ}$ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of $ω_{1}$ null sets in $2^{ω 1}$ such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is “no” for κ = ω. We also give alternative...

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

Hisao Kato (1994)

Fundamenta Mathematicae

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A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that $d (f^{n} (x), f^{n} (y)) > c$ (resp. $d i a m f^{n} (A) > c$ ). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate...

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

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Let $X \subseteq 2^{w}$ . Consider the class of all Borel $F \subseteq X \times 2^{w}$ with null vertical sections $F_{x}$ , x ∈ X. We show that if for all such F and all null Z ⊆ X, $\cup_{x \in Z} F_{x}$ is null, then for all such F, $\cup_{x \in X} F_{x} \neq 2^{w}$ . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].