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...

Minimal Hausdorff (Baire) group topologies of certain groups of transformations naturally occurring in analysis are studied. The results obtained are subsequently applied to show that, e.g., the homeomorphism groups of the rational and of the irrational numbers carry no Polish group topology. In answer to a question of A. S. Kechris it is shown that the group of Borel automorphisms of ℝ cannot be a Polish group either.

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${\left(e_i\right)}_{i\in \mathbb{N}}$; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${\left[e_i\right]}_{i\in A}\ncong {\left[e_i\right]}_{i\in B}$, or for a residual set of infinite subsets A of ℕ, ${\left[e_i\right]}_{i\in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {\left[e_i\right]}_{i\in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder decomposition...

We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].

We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set 𝓐, in the Effros-Borel space of subspaces of C[0,1], of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space Y, with a Schauder basis, that contains isomorphic copies of every space X in the class 𝓐.

We show that if T is an uncountable Polish space, 𝓧 is a metrizable space and f:T→ 𝓧 is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f[T∖M] is a separable space. We also give an example showing that "metrizable" cannot be replaced by "normal".

We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is ${\prod }_2^0$-complete and that the set of Cauchy problems which locally have a unique solution is ${\sum }_3^0$-complete. We prove that the set of Cauchy problems which have a global solution is ${\sum }_0^4$-complete...

We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w\times 2^w$ and continuous functions $e,f:w^w\to w^w$ such that  • N is $G_{\delta }$ and $N_x:x\in w^w$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_{\sigma }$ and $M_x:x\in w^w$ is a basis for the ideal of meager subsets of $2^w$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq N_y\Rightarrow M_x\subseteq M_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_{\delta }$) sets $B\subseteq X\times 2^w$ with all...

Three sets occurring in functional analysis are shown to be of class PCA (also called ${\Sigma }_2^1$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].