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

We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no perfect set P such that |(eta+B) cap (nu+B)|>5 for any distinct eta,nu from P. This answers two questions from our previous works.

We give several topological/combinatorial conditions that, for a filter on $\omega$, are equivalent to being a non-meager $\mathsf{P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager $\mathsf{P}$-filter. Here, we identify a filter with a subspace of $2^\omega$ through characteristic functions. Along the way, we generalize to non-meager $\mathsf{P}$-filters a result of Miller about $\mathsf{P}$-points, and we employ...

In this article we will investigate nonmeasurability with respect to some $\sigma$-ideals in Polish space $X,$ of images of subsets of $X$ by selected mappings defined on the space $X$. Among of them we answer the following question: "It is true that there exists a subset of the unit disc in the real plane such that the continuum many projections onto lines are Lebesgue measurable and continuum many projections are not?". It is known that there exists continuous function $f:[...

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $\pi$-weight, if ${}^\omega{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.

In [arXiv:1605.02261] Ros\l{}anowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with $ZFC$ that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper we give affirmative answers for the general case.

Given an equivalence class $[A]$ in the measure algebra of the Cantor space, let $\hat\Phi([A])$ be the set of points having density 1 in $A$. Sets of the form $\hat\Phi([A])$ are called $\mathcal{T}$-regular. We establish several results about $\mathcal{T}$-regular sets. Among these, we show that $\mathcal{T}$-regular sets can have any complexity within $\Pi^{0}_{3}$ (=$ \mathbf{F}_{\sigma\delta}$), that is for any $\Pi^{0}_{3}$ subset $X$ of the Cantor space there is a $\ma...

The existence of an uncountable family of nonmeager filter whose intersection is meager is consistent with MA(Suslin)

In this note we consider an arbitrary families of sets of $s_0$ ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space $X$ and under some combinatorial and set theoretical assumptions ($cov(s_0)=\c$ for example), that for any family $\ca\subseteq s_0$ with $\bigcup\ca =X$, we can find a some subfamily $\ca'\subseteq\ca$ such that the union $\bigcup\ca'$ is not $s$-measurable. We have shown a consistency of the $cov(s_0)=\omega_1<\c$ and existen...

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.

We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszy\'nski and Judah.