Perfect set theorems

Otmar Spinas

Displaying similar documents to “Perfect set theorems”

The $G_{δ}$ -topology and K-analytic spaces without perfect compact sets

Jose L. Blasco (1990)

Colloquium Mathematicae

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Factors of a perfect square

Tsz Ho Chan (2014)

Acta Arithmetica

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We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between $\sqrt n - ∜ n {(l o g n)}^{1 / 7}$ and $\sqrt n + ∜ n {(l o g n)}^{1 / 7}$ .

Finite-to-one continuous s-covering mappings

Alexey Ostrovsky (2007)

Fundamenta Mathematicae

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The following theorem is proved. Let f: X → Y be a finite-to-one map such that the restriction $f | f^{- 1} (S)$ is an inductively perfect map for every countable compact set S ⊂ Y. Then Y is a countable union of closed subsets $Y_{i}$ such that every restriction $f | f^{- 1} (Y_{i})$ is an inductively perfect map.

Perfect set properties in models of ZF

Carlos Augusto Di Prisco, Franklin C. Galindo (2010)

Fundamenta Mathematicae

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We study several perfect set properties of the Baire space which follow from the Ramsey property $ω \to {(ω)}^{ω}$ . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

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...

An independency result in connectification theory

Alessandro Fedeli, Attilio Le Donne (1999)

Commentationes Mathematicae Universitatis Carolinae

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A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let $ψ$ be the following statement: “a perfect $T_{3}$ -space $X$ with no more than $2^{𝔠}$ clopen subsets is connectifiable if and only if no proper nonempty clopen subset of $X$ is feebly compact". In this note we show that neither $ψ$ nor $\neg ψ$ is provable in ZFC.

Less than $2^{ω}$ many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

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We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $c o f () < 2^{ω}$ ) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ.

Possibly there is no uniformly completely Ramsey null set of size $2^{ω}$

Andrzej Nowik (2002)

Colloquium Mathematicae

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We show that under the axiom $C P A_{c u b e}$ there is no uniformly completely Ramsey null set of size $2^{ω}$ . In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

A note on perfect matchings in uniform hypergraphs with large minimum collective degree

Vojtěch Rödl, Andrzej Ruciński, Mathias Schacht, Endre Szemerédi (2008)

Commentationes Mathematicae Universitatis Carolinae

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For an integer $k \geq 2$ and a $k$ -uniform hypergraph $H$ , let $δ_{k - 1} (H)$ be the largest integer $d$ such that every $(k - 1)$ -element set of vertices of $H$ belongs to at least $d$ edges of $H$ . Further, let $t (k, n)$ be the smallest integer $t$ such that every $k$ -uniform hypergraph on $n$ vertices and with $δ_{k - 1} (H) \geq t$ contains a perfect matching. The parameter $t (k, n)$ has been completely determined for all $k$ and large $n$ divisible by $k$ by Rödl, Ruci’nski, and Szemerédi in [, submitted]. The values of $t (k, n)$ are very close to $n / 2 - k$ . In fact, the function $t (k, n) = n / 2 - k + c_{n, k}$ ,...

Quasiperfect domination in triangular lattices

Italo J. Dejter (2009)

Discussiones Mathematicae Graph Theory

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A vertex subset S of a graph G is a perfect (resp. quasiperfect) dominating set in G if each vertex v of G∖S is adjacent to only one vertex ( $d_{v}$ ∈ 1,2 vertices) of S. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol 3,6 and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets S with induced components of the form $K_{ν}$ , where ν ∈ 1,2,3 depends only...

On the index of an odd perfect number

Feng-Juan Chen, Yong-Gao Chen (2014)

Colloquium Mathematicae

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Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} | | N$ . Define the index $m = σ (N / q^{α}) / q^{α}$ . We prove that m cannot take the form $p^{2 u}$ , where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅,...

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

Michał Morayne (1987)

Colloquium Mathematicae

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Compact covering mappings and cofinal families of compact subsets of a Borel set

G. Debs, J. Saint Raymond (2001)

Fundamenta Mathematicae

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Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement " $\forall α \in ω^{ω}, ω ₁^{L (α)} < ω ₁$ "; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets...

Bérenger/Maxwell with Discontinous Absorptions : Existence, Perfection, and No Loss

Laurence Halpern, Jeffrey Rauch (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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We analyse Bérenger’s split algorithm applied to the system version of the two dimensional wave equation with absorptions equal to Heaviside functions of $x_{j}$ , $j = 1, 2$ . The methods form the core of the analysis [11] for three dimensional Maxwell equations with absorptions not necessarily piecewise constant. The split problem is well posed, has no loss of derivatives (for divergence free data in the case of Maxwell), and is perfectly matched.

Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of $C^{n}$

J. Siciak (1969)

Annales Polonici Mathematici

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The analytic $α$ -capacity and the Cauchy integral

K. Adžievski (1986)

Matematički Vesnik

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