Let $(X,𝕀)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R\left(A^2\right)$ is completely $𝕀$-nonmeasurable, i.e, it is $𝕀$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations ${\left(R_{\alpha }\right)}_{\alpha <2^{\omega }}.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.

We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers ℙ not $G_{\delta }·F_{\sigma }$ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ ℙ of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. The answer to the question as stated is negative if X is Baire but Y is not Baire.

We investigate some geometrical properties of squares of special Sierpiński sets. In particular, we prove that (under CH) there exists a Sierpiński set S and a function p: S → S such that the images of the graph of this function under π'(⟨x,y⟩) = x - y and π''(⟨x,y⟩) = x + y are both Lusin sets.

We study the generalized Cantor space ${}^{\kappa }2$ and the generalized Baire space ${}^{\kappa }\kappa$ as analogues of the classical Cantor and Baire spaces. We equip ${}^{\kappa }\kappa$ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of ${}^{\kappa }2$. We prove for successor $\kappa ={\kappa }^{<\kappa }$ that the ideal of strong measure zero sets of ${}^{\kappa }2$ is ${}_{\kappa }$-additive, where ${}_{\kappa }$ is the size of the smallest unbounded family in ${}^{\kappa }\kappa$, and that the generalized Borel conjecture...

The cardinal invariants $𝔥,𝔟,𝔰$ of $𝒫\left(\omega \right)$ are known to satisfy that ${\omega }_1\le 𝔥\le min\{𝔟,𝔰\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $𝔥$ and show that it can be less than $𝔰$. The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{\omega }$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{\omega }$. We show that $S_{\omega }$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...

Let $M$ be the set of all Dirichlet measures on the unit circle. We prove that $M+M$ is a non Borel analytic set for the weak* topology and that $M+M$ is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates $M+M$ from $D^{\perp }$ (or even $L_0^{\perp })$, the set of all measures singular with respect to every measure in $M$. This extends results of Kaufman, Kechris and Lyons about $D^{\perp }$ and $H^{\perp }$ and gives many examples of non Borel analytic sets.

If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.

We show that the first homology group of a locally connected compact metric space is either uncountable or finitely generated. This is related to Shelah's well-known result (1988) which shows that the fundamental group of such a space satisfies a similar condition. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result does not follow from Shelah's. We clarify a claim made by Pawlikowski (1998) and offer a proof of the...