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We use Tsirelson’s Banach space ([2]) to define an P-ideal which refutes a conjecture of Mazur and Kechris (see [12, 9, 8]).
For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces , and such that
(i) , where f is either trdef or ₀-trsur,
(ii) and ,
(iii) and , and
(iv) and .
We also show that there exists no separable metrizable space with , and , where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.
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.
We construct in Bell-Kunen’s model: (a) a group maximal topology on a countable infinite Boolean group of weight and (b) a countable irresolvable dense subspace of . In this model .
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