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The Banach–Mazur game and σ-porosity

Miroslav Zelený — 1996

Fundamenta Mathematicae

It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.

Sets of extended uniqueness and σ -porosity

Miroslav Zelený — 1997

Commentationes Mathematicae Universitatis Carolinae

We show that there exists a closed non- σ -porous set of extended uniqueness. We also give a new proof of Lyons’ theorem, which shows that the class of H ( n ) -sets is not large in U 0 .

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

P. HolickýMiroslav Zelený — 2000

Fundamenta Mathematicae

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 image of...

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

Miroslav ZelenýLuděk 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.

Rudin-like sets and hereditary families of compact sets

Étienne MatheronMiroslav Zelený — 2005

Fundamenta Mathematicae

We show that a comeager Π₁¹ hereditary family of compact sets must have a dense G δ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true F σ δ sets.

On the complexity of some σ -ideals of σ -P-porous sets

Luděk ZajíčekMiroslav Zelený — 2003

Commentationes Mathematicae Universitatis Carolinae

Let 𝐏 be a porosity-like relation on a separable locally compact metric space E . We show that the σ -ideal of compact σ - 𝐏 -porous subsets of E (under some general conditions on 𝐏 and E ) forms a Π 1 1 -complete set in the hyperspace of all compact subsets of E , in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the σ -ideals of σ -porous sets, σ - g -porous sets, σ -strongly porous sets, σ -symmetrically porous sets...

The structure of the σ -ideal of σ -porous sets

Miroslav ZelenýJan Pelant — 2004

Commentationes Mathematicae Universitatis Carolinae

We show a general method of construction of non- σ -porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non- σ -porous Suslin subset of a topologically complete metric space contains a non- σ -porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non- σ -porous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Vietoris topology...

A remark on a theorem of Solecki

Petr HolickýLuděk ZajíčekMiroslav Zelený — 2005

Commentationes Mathematicae Universitatis Carolinae

S. Solecki proved that if is a system of closed subsets of a complete separable metric space X , then each Suslin set S X which cannot be covered by countably many members of contains a G δ set which cannot be covered by countably many members of . We show that the assumption of separability of X cannot be removed from this theorem. On the other hand it can be removed under an extra assumption that the σ -ideal generated by is locally determined. Using Solecki’s arguments, our result can be used...

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