Eleftherios Tachtsis

Compact Metric Spaces and Weak Forms of the Axiom of Choice

It is shown that for compact metric spaces (X, d) the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn∞ diam (Gn) = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”.

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well-ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω).

On Russell and anti Russell-cardinals

Abstract In the absence of the Axiom of Choice we study countable families of 2-element sets with no choice functions which either have infinite subfamilies with a choice function or no infinite subfamilies with a choice function.

Countable compact Hausdorff spaces need not be metrizable in ZF

We show that the existence of a countable, first countable, zero-dimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF.

Countable sums and products of metrizable spaces in ZF

We study the role that the axiom of choice plays in Tychonoffs product theorem restricted to countable families of compact, as well as, Lindelof metric spaces, and in disjoint topological unions of countably many such spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On Sequentially Compact Subspaces of without the Axiom of Choice

We show that the property of sequential compactness for subspaces of R is countably productive in ZF. Also, in the language of weak choice principles, we give a list of characterizations of the topological statement ‘sequentially compact subspaces of R are compact’. Furthermore, we show that forms 152 (= every non-well-orderable set is the union of a pairwise disjoint well-orderable family of denumerable sets) and 214 (= for every family A of infinite sets there is a function f such that for all y ∈ A, f (y) is a nonempty subset of y and | f (y)| = א0) of Howard and Rubin are equivalent.

The cardinal inequality α2 < 2α

Abstract In ZFC set theory (i.e., Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) any two cardinal numbers are comparable. However, this may not be valid in ZF (i.e., Zermelo-Fraenkel set theory modulo AC). In this paper, we study the strength of the inequality α2 < 2α (in ZF, for every infinite cardinal number α, 2α ≰ α2; see [13]), where α is either the cardinality of special sets (see Definition 1 below) which are expressed as disjoint unions of finite, pairwise equipotent, sets lacking choice functions, or the cardinality of sets in specific permutation models of ZF0 set theory (i.e., ZF without the Axiom of Regularity).

Weak axioms of choice for metric spaces

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space (X, d) has a choice function, then so does the family of all non-empty, open subsets of X. In addition, we establish that the converse is not provable in ZF. We also show that the statement every subspace of the real line R with the standard topology has a choice function for its family of all closed, non-empty subsets is equivalent to the weak choice form every continuum sized family of non-empty subsets of reals has a choice function.

Choice principles for special subsets of the real line

We study the role the axiom of choice plays in the existence of some special subsets of ℝ and its power set ℘(ℝ).

On the minimal cover property and certain notions of finite

In set theory without the axiom of choice, we investigate the deductive strength of the principle “every topological space with the minimal cover property is compact”, and its relationship with certain notions of finite as well as with properties of linearly ordered sets and partially ordered sets.