Very small sets

Haim Judah; Amiran Lior; Ireneusz Recław

Displaying similar documents to “Very small sets”

Generalized projections of Borel and analytic sets

Marek Balcerzak (1996)

Colloquium Mathematicae

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For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^{2}$ , we consider the generalized projection (A) of A given by (A) = x ∈ X: Ax ∉ I, where $A_{x}$ =y ∈ X: 〈x,y〉∈ A. We study the behaviour of with respect to Borel and analytic sets in the case when I is a $\sum_{2}^{0}$ -supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that [ $\sum_{1}^{1} (X^{2})] = \sum_{1}^{1} (X)$ for a wide class of $\sum_{2}^{0}$ -supported σ-ideals.

Mycielski ideals generated by uncountable systems

A. Rosłanowski (1993)

Colloquium Mathematicae

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A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections

P. Holický, Miroslav Zelený (2000)

Fundamenta Mathematicae

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Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then $f^{- 1} (y)$ is a $K_{σ}$ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the...

Some additive properties of special sets of reals

Ireneusz Recław (1991)

Colloquium Mathematicae

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Borel ideals in three variables.

Marinari, Maria Grazia, Ramella, Luciana (2006)

Beiträge zur Algebra und Geometrie

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Convergence with respect to $F_{σ}$ -supported ideals

Jacek Hejduk (1997)

Colloquium Mathematicae

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Additive properties and uniformly completely Ramsey sets

Andrzej Nowik (1999)

Colloquium Mathematicae

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We prove some properties of uniformly completely Ramsey null sets (for example, every hereditarily Menger set is uniformly completely Ramsey null).

Vitali sets and Hamel bases that are Marczewski measurable

Arnold Miller, Strashimir Popvassilev (2000)

Fundamenta Mathematicae

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We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals...

Uniformly completely Ramsey sets

Udayan Darji (1993)

Colloquium Mathematicae

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Galvin and Prikry defined completely Ramsey sets and showed that the class of completely Ramsey sets forms a σ-algebra containing open sets. However, they used two definitions of completely Ramsey. We show that they are not equivalent as they remarked. One of these definitions is a more uniform property than the other. We call it the uniformly completely Ramsey property. We show that some of the results of Ellentuck, Silver, Brown and Aniszczyk concerning completely Ramsey sets also...