Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set

A set of reals A is called perfectly meager if A \cap P is meager in P, for every perfect set P. Marczewski asked if the product of perfectly meager sets is perfectly meager. In the paper it is shown that it is consistent that the answer to this question is positive. (It is known that it is also consistent that the answer is negative (Reclaw))

In this article we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff-measure is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set satisfies that there exists a translation-invariant measure mu for which the set has positive and finite mu-measure. In contrast, we generalize an example of Davies of dimensionles...

We study a strengthening of the notion of a perfectly meager set. We say that that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager in $X$, if for every sequence of perfect subsets $\{P_n: n \in {\mathbb N}\}$ of $X$, there exists an $F_\sigma$-set $F$ in $X$ such that $A \subseteq F$ and $F\cap P_n$ is meager in $P_n$ for each $n$. We give various characterizations and examples of countably perfectly meager sets. We prove that not every universall...

Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points $x$ such that the density of $x$ at $K$ exists and it is different from $0$ or $1$ is $\Pi^{0}_{3}$-complete; the set of all $[K]$ with $K$ compact is $\Pi^{0}_{3}$-complete. There is a set (which can be taken to be open or closed) in $\mat...

The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article we will explore the richness, the peculiarities and the generalities that has $C$ and explore some variants and generalizations of it. For a more systematic treatment the Cantor like sets we refer to our previous paper.

The paper contains two results pointing to the lack of symmetry between measure and category. Assume CH. There exists a strongly meager subset of the Cantor set that can be mapped onto the Cantor set by a uniformly continuous function. (It is well known that uniformly continuous image of a strongly null set is strongly null). A ZFC version of this result is also given.

We construct a combinatorially large measure zero subset of the Cantor set.

We prove that for each meager relation $E\subset X\times X$ on a Polish space $X$ there is a nowhere meager subspace $F\subset X$ which is $E$-free in the sense that $(x,y)\notin E$ for any distinct points $x,y\in F$.

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leq\omega$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such that $F$ is $R$-free for every $R\in\mathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $\omega$, and define the corresponding cardinal invar...