Bounded countable atomic compactness of ordered groups

F U N D a M E N T a Mathematicae Bounded Countable Atomic Compactness of Ordered Groups

We show that whenever A is a monotone σ-complete dimension group, then A+ ∪ {∞} is countably equationally compact, and we show how this property can supply the necessary amount of completeness in several kinds of problems. In particular, if A is a countable dimension group and E is a monotone σ-complete dimension group, then the ordered group of all relatively bounded homomorphisms from A to E ...

Absolute countable compactness of products and topological groups

In this paper, we generalize Vaughan’s and Bonanzinga’s results on absolute countable compactness of product spaces and give an example of a separable, countably compact, topological group which is not absolutely countably compact. The example answers questions of Matveev [8, Question 1] and Vaughan [9, Question (1)].

Countable Choice and Compactness

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p ≥ 1 (resp. p = 0), and some closed subse...

PI11-CA0 and Order Types of Countable Ordered Groups

Compactness in Countable Tychonoff Products and Choice

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.