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.

We construct a differentiable function $f:{\mathbf{R}}^{n}\to \mathbf{R}$ ($n\ge 2$) such that the set ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.

We show that there exists a closed non-$\sigma $-porous set of extended uniqueness. We also give a new proof of Lyons’ theorem, which shows that the class of ${H}^{\left(n\right)}$-sets is not large in ${U}_{0}$.

Let $X={\ell}_{p}$, $p\in (2,+\infty )$. We construct a function $f:X\to \mathbb{R}$ which has Lipschitz Fréchet derivative on $X$ but is not a d.c. function.

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 ${f}^{-1}\left(y\right)$ is a ${K}_{\sigma}$ 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...

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 show that in every Polish, abelian, non-locally compact group G there exist non-Haar null sets A and B such that the set {g ∈ G; (g+A) ∩ B is non-Haar null} is empty. This answers a question posed by Christensen.

We construct, for each countable ordinal ξ, a closed graph with Borel chromatic number 2 and Baire class ξ chromatic number ℵ₀.

We show that a comeager Π₁¹ hereditary family of compact sets must have a dense ${G}_{\delta}$ 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 ${F}_{\sigma \delta}$ sets.

Let $\mathbf{P}$ be a porosity-like relation on a separable locally compact metric space $E$. We show that the $\sigma $-ideal of compact $\sigma $-$\mathbf{P}$-porous subsets of $E$ (under some general conditions on $\mathbf{P}$ and $E$) forms a ${\Pi}_{\mathbf{1}}^{\mathbf{1}}$-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 $\sigma $-ideals of $\sigma $-porous sets, $\sigma $-$\langle g\rangle $-porous sets, $\sigma $-strongly porous sets, $\sigma $-symmetrically porous sets...

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

S. Solecki proved that if $\mathcal{F}$ is a system of closed subsets of a complete separable metric space $X$, then each Suslin set $S\subset X$ which cannot be covered by countably many members of $\mathcal{F}$ contains a ${G}_{\delta}$ set which cannot be covered by countably many members of $\mathcal{F}$. 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 $\sigma $-ideal generated by $\mathcal{F}$ is locally determined. Using Solecki’s arguments, our result can be used...

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