We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.

We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of ${\omega }^{\omega }$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

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.

A subobjects structure of the category $\Omega$- of $\Omega$-fuzzy sets over a complete $MV$-algebra $\Omega =(L,\wedge ,\vee ,\otimes ,\to )$ is investigated, where an $\Omega$-fuzzy set is a pair $𝐀=(A,\delta )$ such that $A$ is a set and $\delta A\times A\to \Omega$ is a special map. Special subobjects (called complete) of an $\Omega$-fuzzy set $𝐀$ which can be identified with some characteristic morphisms $𝐀\to {\Omega }^{*}=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms ${\neg }_{\Omega }{\Omega }^{*}\to {\Omega }^{*},{\cap }_{\Omega }$, ${\cup }_{\Omega }{\Omega }^{*}\times {\Omega }^{*}\to {\Omega }^{*}$ are characteristic morphisms of complete subobjects.

Assume that no cardinal κ < 2ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover ⊆ $$𝕀$$ of X contains uncountably many pairwise disjoint subfamilies , with $$𝕀$$ -Bernstein unions ∪ (a subset A ⊆ X is $$𝕀$$ -Bernstein if A and X A meet each Borel $$𝕀$$ -positive subset...

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.

If T is a complete theory stronger than ZF${}_{\text{Fin}}$ such that axiom of extensionality for classes + T + $(\exists X){\Phi }_i$ is consistent for 1$\le i\le k$ (each alone), where ${\Phi }_i$ are normal formulae then we show AST + $(\exists X){\Phi }_1+\cdots +(\exists X){\Phi }_k$ + scheme of choice is consistent. As a consequence we get: there is no proper ${\Delta }_1$-formula in AST + scheme of choice. Moreover the complexity of the axioms of AST is studied, e.gẇe show axiom of extensionality is ${\Pi }_1$-formula, but not ${\Sigma }_1$-formula and furthermore prolongation axiom, axioms of choice and cardinalities are ${\Pi }_2$-formulae,...

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