We prove some closed mapping theorems on k-spaces with point-countable k-networks. One of them generalizes Lašnev’s theorem. We also construct an example of a Hausdorff space Ur with a countable base that admits a closed map onto metric space which is not compact-covering. Another our result says that a k-space X with a point-countable k-network admitting a closed surjection which is not compac...

The basic theorem presented shows that the product of a linearly ordered space and a countable (regular) space is normal. We prove that the countable space can be replaced by any of a rather large class of countably tight spaces. Examples are given to prove that monotone normality cannot replace linearly ordered in the base theorem. However, it is shown that the product of a monotonically norma...

We prove that the natural order on the idempotents of the endomorphism monoid of the countably in®nite random graph is d 0-universal; that is, it embeds every countable order. We therefore extend in a strong fashion a result of [2] which showed that the natural order embeds every countable linear order. We consider a re®nement of the natural order which embeds every countable quasi-order.

Consider a standard probability space (X,μ), i.e., a space isomorphic to the unit interval with Lebesgue measure. We denote by Aut(X,μ) the automorphism group of (X,μ), i.e., the group of all Borel automorphisms of X which preserve μ (where two such automorphisms are identified if they are equal μ-a.e.). A Borel equivalence relation E ⊆ X is called countable if every E-class [x]E is countable a...

In this paper a simple proof of the completeness theorem of the intuitionistic predicate calculus with respect to the topological semantics is shown. From a technical point of view the proof of the completeness theorem is based on a Rasiowa-Sikirski-like theorem for the countable Heyting algebras which allows to embadd any countable Heyting algebra into a suitable topology in a such way that a ...

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...

We give a classification of all the countable homogeneous multipartite graphs. This generalizes the similar result for bipartite graphs given in [5]. 2010 Mathematics Subject Classification 05C99 keywords: multipartite graph, homogeneous, classification

Introduction, Let © be the class of all countable and connected perfectly separable Hausdorff spaces containing more than one point. I t is known that an ©-space cannot be regular or compact. Urysohn, using a complicated identification of points, has constructed the first example of an ©-space. Two ©-spaces, X and X*, more simply constructed and not involving identifications, are presented here...

This is a revised and extended version of an article which encapsulates a key aspect of the “Henkin method” in a general result about the existence of finitely consistent theories satisfying prescribed closure conditions. This principle can be used to give streamlined proofs of completeness for logical systems, in which inductive Henkin-style constructions are replaced by a demonstration that a...

Region Connection Calculus (RCC) is the most widely studied formalism of Qualitative Spatial Reasoning. It has been known for some time that each connected regular topological space provides an RCC model. These ‘standard’ models are inevitable uncountable and regions there cannot be represented finitely. This paper, however, draws researchers’ attention to RCC models that can be constructed fro...