The article is continuation of [2] and [1], and the goal of it is show that Zermelo theorem (every set has a relation which well orders it proposition (26)) and axiom of choice (for every non-empty family of non-empty and separate sets there is set which has exactly one common element with arbitrary family member proposition (27)) are true. It is result of the Tarski’s axiom A introduced in [5]...

We give here the first examples of C structurally stable maps in manifolds of dimension greater than one that are not diffeomorphisms nor expanding. It is shown that an Axiom A endomorphism all of whose basic pieces are expanding or attracting is C stable. A necessary condition for the existence of such examples is also given.

We revise our ‘Physical Traces’ paper [Abramsky and Coecke CTCS‘02] in the light of the results in [Abramsky and Coecke LiCS‘04]. The key fact is that the notion of a strongly compact closed category allows abstract notions of adjoint, bipartite projector and inner product to be defined, and their key properties to be proved. In this paper we improve on the definition of strong compact closure ...

The power and usefulness of cleavability (also known as splittability) have been well established within the framework of topology by A.V. Arhangel’skii and his associates. The concept was transferred to partially ordered sets by D.J. Marron and T.B.M. McMaster. The connections between topology and order are here exploited as we examine the properties enjoyed by a topological space that is spli...

The main purpose of this paper is to investigate constructively the relationship between proximal convergence, uniform sequential convergence and uniform convergence for sequences of mappings between apartness spaces. It is also shown that if the second space satisfies the Efremovic axiom, then proximal convergence preserves strong continuity. Mathematics Subject Classification: 54E05, 03F60

An Open Coloring Axiom type principle is formulated for uncountable cardinals and is shown to be a consequence of the Proper Forcing Axiom. Several applications are found. We also study dense C∗-embedded subspaces of ω∗, showing that there can be such sets of cardinality c and that it is consistent that ω∗ \ {p} is C∗-embedded for some but not all p ∈ ω∗.