Two applications of analytic topologies.
Two cases when and are equal
We deal with two cardinal invariants and give conditions on their equality using Shelah's pcf theory.
Two constructions of compatible relations
Two counterexamples concerning Hausdorff dimensions of projections
Two dichotomy theorems on colourability of non-analytic graphs
We prove: Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a κ-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph into G. Theorem 2. In the Solovay model, every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above.
Two model theoretic ideas in independence proofs
Two point sets with additional properties
A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some -ideal, being (completely) nonmeasurable with respect to different -ideals, being a -covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations...
Two Problems of Measurability Concerning Convex Sets in Euclidean Spaces.
Two theorems of functional analysis effectively equivalent to choice axioms
Tychonoff products of super second countable and super separable metric spaces
Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem
Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product is compact. (iv) In a Boolean algebra...
Tychonoff's theorem without the axiom of choice
Type d'ordre des ordinaux de modèles non dénombrables de la théorie des ensembles