We propose and study a “classification” of Borel ideals based on a natural infinite game involving a pair of ideals. The game induces a pre-order $⊑$ and the corresponding equivalence relation. The pre-order is well founded and “almost linear”. We concentrate on $F_{\sigma }$ and $F_{\sigma \delta }$ ideals. In particular, we show that all $F_{\sigma }$-ideals are $⊑$-equivalent and form the least equivalence class. There is also a least class of non-$F_{\sigma }$ Borel ideals, and there are at least two distinct classes of $F_{\sigma \delta }$ non-$F_{\sigma }$ ideals.

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega }^{\omega }$ there exists a continuous function $f:{\omega }^{\omega }\to X$ such that $f^{-1}\left(C₀\right)=D₀$ and $f^{-1}\left(C₁\right)=D₁$. We give several explicit examples of complete pairs of coanalytic sets.

The notion of a complete sequence of pairwise disjoint coanalytic sets is investigated. Several examples are given and such sequences are characterised under analytic determinacy. The ideas are based on earlier results of Saint Raymond, and generalise them.

On montre que si G est un groupe abélien localment compact non diskret à base dénombrable d'ouverts, alors la famille des fermés de synthèse pour l'algèbre de Fourier A(G) est une partie coanalytique non borélienne de ℱ(G), l'ensemble des fermés de G muni de la structure borélienne d'Effros. On généralise ainsi un résultat connu dans le cas du groupe 𝕋.

Nous donnons, pour chaque niveau de complexité Γ, une caractérisation du type "test d'Hurewicz" des boréliens d'un produit de deux espaces polonais ayant toutes leurs coupes dénombrables ne pouvant pas être rendus Γ par changement des deux topologies polonaises.

We show that each of the classes of hereditarily locally connected, finitely Suslinian, and Suslinian continua is Π₁¹-complete, while the class of regular continua is Π₀⁴-complete.

We evaluate the descriptive set theoretic complexity of the space of continuous surjections from ${ℝ}^m$ to ${ℝ}^n$.

If G is a countable group containing a copy of F₂ then the conjugacy equivalence relation on subgroups of G attains the maximal possible complexity.

Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space ${\kappa }^{\kappa }$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in ${\kappa }^{\kappa }$ for uncountable regular κ is however consistent (with GCH), assuming...

A subset of a Polish space X is called universally small if it belongs to each ccc σ-ideal with Borel base on X. Under CH in each uncountable Abelian Polish group G we construct a universally small subset A₀ ⊂ G such that |A₀ ∩ gA₀| = for each g ∈ G. For each cardinal number κ ∈ [5,⁺] the set A₀ contains a universally small subset A of G with sharp packing index $pack♯\left(A_{\kappa }\right)=sup\left|\right|⁺:\subset {gA}_{g\in G}isdisjoint$ equal to κ.

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear...

Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f: S → ℝ is of the form f(x,y) = u(x) + v(y), where u: X → ℝ and v: Y → ℝ are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures...

Answering a question of Kłopotowski, Nadkarni, Sarbadhikari, and Srivastava, we characterize the Borel sets S ⊆ X × Y with the property that every Borel function f: S → ℂ is of the form f(x,y) = u(x) + v(y), where u: X → ℂ and v: Y → ℂ are Borel.

We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An...

We develop the theory of topological Hurewicz test pairs: a concept which allows us to distinguish the classes of the Borel hierarchy by Baire category in a suitable topology. As an application we show that for every ${\Pi }_{\xi }^0$ and not ${\Sigma }_{\xi }^0$ subset $P$ of a Polish space $X$ there is a $\sigma$-ideal $ℐ\subseteq 2^X$ such that $P\notin ℐ$ but for every ${\Sigma }_{\xi }^0$ set $B\subseteq P$ there is a ${\Pi }_{\xi }^0$ set $B^{\text{'}}\subseteq P$ satisfying $B\subseteq B^{\text{'}}\in ℐ$. We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.