We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.

This note is devoted to combinatorial properties of ideals on the set of natural numbers. By a result of Mathias, two such properties, selectivity and density, in the case of definable ideals, exclude each other. The purpose of this note is to measure the ``distance'' between them with the help of ultrafilter topologies of Louveau.

Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement "$\forall \alpha \in {\omega }^{\omega },\omega {₁}^{L\left(\alpha \right)}<\omega ₁$"; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets of a Borel...

We study homeomorphism groups of metrizable compactifications of ℕ. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_{\infty }$. As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_{\infty }$. We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on ℕ a certain Polishable...

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...