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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.

The covering property for σ-ideals of compact, sets

Carlos Uzcátegui (1992)

Fundamenta Mathematicae

The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of 2 ω .

The Hausdorff dimension of some special plane sets

Jan Mařík (1994)

Mathematica Bohemica

A compact set T 𝐑 2 is constructed such that each horizontal or vertical line intersects T in at most one point while the α -dimensional measure of T is infinite for every α ( 0 , 2 ) .

The measure extension problem for vector lattices

J. D. Maitland Wright (1971)

Annales de l'institut Fourier

Let V be a boundedly σ -complete vector lattice. If each V -valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a σ -additive measure on the generated σ -field then V is said to have the measure extension property. Various sufficient conditions on V which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: V has the...

The prevalence of permutations with infinite cycles

Randall Dougherty, Jan Mycielski (1994)

Fundamenta Mathematicae

A number of recent papers have been devoted to the study of prevalence, a generalization of the property of being of full Haar measure to topological groups which need not have a Haar measure, and the dual concept of shyness. These concepts give a notion of "largeness" which often differs from the category analogue, comeagerness, and may be closer to the intuitive notion of "almost everywhere." In this paper, we consider the group of permutations of natural numbers. Here, in the sense of category,...

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

Miroslav Zelený, Jan 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...

The σ-ideal of closed smooth sets does not have the covering property

Carlos Uzcátegui (1996)

Fundamenta Mathematicae

We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.

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