Rigid Unary Functions and the Axiom of Choice

We shall investigate certain statements concerning the rigidity of unary functions which have connections with (weak) forms of the axiom of choice.     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

The Compactness of 2ℝ and the Axiom of Choice

We show that for every we ordered cardinal number m the Tychonoff product 2^m is a compact space without the use of any choice but in Cohen's Second Mode 2^ℝ is not compact.     

The Vector Space Kinna‐Wagner Principle is Equivalent to the Axiom of Choice

We show that the axiom of choice AC is equivalent to the Vector Space Kinna‐Wagner Principle, i.e., the assertion: “For every family 𝒱= {V_i : i ∈ k} of non trivial vector spaces there is a family ℱ = {F_i : i ∈ k} such that for each i ∈ k, F_i is a non empty independent subset of V_i”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC.     

Destructibility of stationary subsets of Pκλ

For a regular cardinal κ with κ ^<κ = κ and κ ≤ λ , we construct generically (forcing by a < κ‐closed κ ⁺‐c. c. p. o.‐set ℙ₀) a subset S of {x ∈ P _κ λ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense (F ^IA_κ λ ‐stationary in our terminology) but the stationarity of S can be destroyed by a κ ⁺‐c. c. forcing ℙ* (in V ^ℙ) which does not add any new element of P _κ λ . Actually ℙ* can be chosen so that ℙ* is κ‐strategically closed. However we show that such ℙ* itself cannot be κ‐strategically closed or even <κ‐strategically closed if κ is inaccessible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bernstein sets and κ ‐coverings

In this paper we study a notion of a κ ‐covering set in connection with Bernstein sets and other types of non‐measurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The wi‐club filter on 𝒫κ λ

We develop the theory of C_{κ, λ}ⁱ, a strongly normal filter over ??_κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??_κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains C_{κ, λ}ⁱ. We also show that functions can be used to obtain a basis for C_{κ, λ}ⁱ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Weak Forms of the Axiom of Choice Restricted to the Real Line

It is shown that AC(ℝ), the axiom of choice for families of non‐empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2 ${}^{\text{ℵ}{}_{\text{0}}}$ ” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 ${}^{\text{ℵ}{}_{\text{0}}}$ has power 2 ${}^{\text{ℵ}{}_{\text{0}}}$ , then the union has power 2 ${}^{\text{ℵ}{}_{\text{0}}}$ ” and (ii) “ℵ(2 ${}^{\text{ℵ}{}_{\text{0}}}$ ) ≠ ℵ_ω” (ℵ(2 ${}^{\text{ℵ}{}_{\text{0}}}$ ) is Hartogs' aleph, the least ℵ not ≤ 2 ${}^{\text{ℵ}{}_{\text{0}}}$ ), is strictly weaker than the full axiom of choice AC.     

Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

Properties of the real line and weak forms of the Axiom of Choice

We investigate, within the framework of Zermelo‐Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

Filters,Antichains and Towers in Topological Spaces and the Axiom of Choice

We find some characterizations of the Axiom of Choice (AC) in terms of certain families of open sets in T₁ spaces.     

Weak Forms of the Axiom of Choice and the Generalized Continuum Hypothesis

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.     

A partition property of a mixed type for Pκ(λ)

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on P_κ(λ) that satisfy the square brackets partition relation , where μ is a cardinal ≥2. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)