We characterize the class of definable families of countable sets for which there is a single countable definable set intersecting every element of the family.

We show that if ℱ is a hereditary family of subsets of ${\omega }^{\omega }$ satisfying certain definable conditions, then the ${\Delta }_1^1$ reals are precisely the reals α such that $\beta :\alpha \in {\Delta }_1^1\left(\beta \right)\notin \in ℱ$. This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the $Q_{2n+1}$-encodable reals is also shown.

We prove that density preserving homeomorphisms form a Π11-complete subset in the Polish space ℍ of all increasing autohomeomorphisms of unit interval.

We relate some subsets $G$ of the product $X\times Y$ of nonseparable Luzin (e.g., completely metrizable) spaces to subsets $H$ of ${\mathbb{N}}^{\mathbb{N}}\times Y$ in a way which allows to deduce descriptive properties of $G$ from corresponding theorems on $H$. As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points $y$ in $Y$ with particular properties of fibres $f^{-1}\left(y\right)$ of a mapping $fX\to Y$. Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.

Let $$𝒦$$ (ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that $$\{K\in 𝒦\left(ℝ\right):{\forall }_x\in K(d^{+}(x,K)=1or{d}^{-}(x,K)=1)\}$$ is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.

We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions ${\Delta }_{h}f\left(x\right)=f(x+h)-f\left(x\right)$ are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, $ℌ(ℱ,G)=H\subset ℝ/\mathbb{Z}:\left(\exists f\in ℱG\right)\left(\forall h\in H\right){\Delta }_{h}f\in G$, we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group $𝕋=ℝ/\mathbb{Z}$ that are invariant for changes on null-sets (e.g. measurable...