we show that in {bf zf} set theory without choice, the ultrafilter mbox{principle} ({bf up}) is equivalent to several compactness theorems for alexandroff discrete spaces and to rudin's lemma, a basic tool in topology and the theory of quasi-continuous domains. important consequences of rudin's lemma are various lift lemmas, saying that certain properties of posets are inherited by th...

Eighty years ago, Felix Hausdorff and Paul Alexandroff published independently a theorem asserting that every compact metric space is a continuous image of the Cantor set. This theorem found its application in various branches of mathematics and played also an important role in the theory of curves. The complete characterization of continuous interval images (i.e. Jordan curves) given by the Ha...

Galois connection in category theory play an important role inestablish the relationships between different spatial structures. Inthis paper, we prove that there exist many interesting Galoisconnections between the category of Alexandroff $L$-fuzzytopological spaces, the category of reflexive $L$-fuzzyapproximation spaces and the category of Alexandroff $L$-fuzzyinterior (closure) spaces. This ...

If X is a topological space, then we let H(X) denote the group of autohomeomorphisms of X equipped with the compact-open topology. For subsets A and B of X we define [A, B] = {h ∈ H(X) : h(A) ⊂ B}, and we recall that the topology on H(X) is generated by the subbasis SX = {[K , O] : K compact, O open in X}. If X is a compact Hausdorff space, then H(X) is a topological group, that is, composition...

A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on -terms closed under and -conversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce a new technique to prove the incompleteness of a wide range of lambda calculus semant...

This paper deals with a thinning algorithm proposed in 2001 by Kovalevsky, for 2D binary images modelled by cell complexes, or, equivalently, by Alexandroff T0 spaces. We apply the general proposal of Kovalevsky to cell complexes corresponding to the three possible normal tilings of congruent convex polygons in the plane: the quadratic, the triangular, and the hexagonal tilings. For this case, ...

This paper provides a theoretical foundation of a thinning method due to Kovalevsky for 2D digital binary images modelled by cell complexes or, equivalently, by Alexandroff T0 topological spaces, whenever these are constructed from polygonal tilings. We analyze the relation between local and global simplicity of cells, and prove their equivalence under certain conditions. For the proof we apply...

The notion of a bisimulation relation is of basic importance in many areas of computation theory and logic. Of late, it has come to take a particular significance in work on the formal analysis and verification of hybrid control systems, where system properties are expressible by formulas of the modal μ-calculus or weaker temporal logics. Our purpose here is to give an analysis of the concept o...