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Ideals induced by Tsirelson submeasures

Ilijas Farah (1999)

Fundamenta Mathematicae

We use Tsirelson’s Banach space ([2]) to define an F σ P-ideal which refutes a conjecture of Mazur and Kechris (see [12, 9, 8]).

Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko, Yasunao Hattori (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces X α , Y α and Z α such that (i) f X α , f Y α , f Z α = ω , where f is either trdef or ₀-trsur, (ii) A ( α ) - t r i n d X α = and M ( α ) - t r i n d X α = - 1 , (iii) A ( α ) - t r i n d Y α = - 1 and M ( α ) - t r i n d Y α = , and (iv) A ( α ) - t r i n d Z α = M ( α ) - t r i n d Z α = and A ( α + 1 ) M ( α + 1 ) - t r i n d Z α = - 1 . We also show that there exists no separable metrizable space W α with A ( α ) - t r i n d W α , M ( α ) - t r i n d W α and A ( α ) M ( α ) - t r i n d W α = , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

Inscribing compact non-σ-porous sets into analytic non-σ-porous sets

Miroslav Zelený, Luděk Zajíček (2005)

Fundamenta Mathematicae

The main aim of this paper is to give a simpler proof of the following assertion. Let A be an analytic non-σ-porous subset of a locally compact metric space, E. Then there exists a compact non-σ-porous subset of A. Moreover, we prove the above assertion also for σ-P-porous sets, where P is a porosity-like relation on E satisfying some additional conditions. Our result covers σ-⟨g⟩-porous sets, σ-porous sets, and σ-symmetrically porous sets.

Irresolvable countable spaces of weight less than

Viacheslav I. Malykhin (1999)

Commentationes Mathematicae Universitatis Carolinae

We construct in Bell-Kunen’s model: (a) a group maximal topology on a countable infinite Boolean group of weight 1 < and (b) a countable irresolvable dense subspace of 2 ω 1 . In this model = ω 1 .

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