On countable stable structures which are homogeneous for a finite relational language

LetL be a finite relational language andH(L) denote the class of all countable structures which are stable and homogeneous forL in the sense of Fraissé. By convention countable includes finite and any finite structure is stable. A rank functionr :H(L) →ω is introduced and also a notion of dimension for structures inH(L). A canonical way of shrinking structures is defined which reduces their dimensions. The main result of the paper is that anyM ∈H(L) can be shrunk toM′ ∈H(L),M′ ⊆M, such that |M′| is bounded in terms ofr(M), and the isomorphism type ofM overM′ is uniquely determined by the dimensions ofM. Forr<ω we deduce thatH(L, r), the class of allM ∈H(L) withr(M)≦r, is the union of a finite number of classes within each of which the isomorphism type of a structure is completely determined by its dimensions. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death     

Hamilton cycles in line graphs of 3-hypergraphs

We prove that every 52-connected line graph of a rank 3 hypergraph is Hamiltonian. This is the first result of this type for hypergraphs of bounded rank other than ordinary graphs.     

The conjugacy problem for automorphism groups of countable homogeneous structures

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.     

On better quasi-ordering countable trees

The main result is that the class of countable trees is better-quasi-ordered under embeddability. R. Laver proved before that a certain class of well-founded trees is b.q.o. Actually our better-quasi-ordered class is larger than the countable class but does not contain Suslin like-trees nor certain Galvin trees of height ω + 1.     

On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {E_n} such that for every positive integer n, the topological dual space of E_n, E′_n, provided with the Mackey topology μ(E′_n,E_n), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.     

On the countable definability of sets

LetQ be a connected set in ^p . Denote byD[Q] the set of all domains containingQ, and letW(Q) be the set of all convex domains fromD[Q]. We present tests for classesD[Q] andW(Q) (in the case whenQ is convex for the last one) to have a countable basis. The results are expressed in terms of properties of the boundary FrQ of the setQ.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 382–395, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-242.     

On generating countable sets of endomorphisms

Sierpiski proved that every countable set of mappings on an infinite set X is contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this paper we prove that every countable set of endomorphisms of an algebra which has an infinite basis (independent generating set) is contained in a 2-generated subsemigroup of the semigroup of all endomorphisms of .     

On the contrapositive of countable choice

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for ${\Sigma_{2}^{0}}$ -formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.     

On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet‐Urysohn space. Although, in its absence even ? may fail to be a sequential space. Our goal in this paper is to discuss under which set‐theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ?, are classes of Fréchet‐Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ? if and only if the axiom of countable choice holds for families of subsets of ?, and every metric space has a unique ‐completion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ? X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω ₁-monolithic compact space X, if C _p (X)is star countable then it is Lindelöf.     

On countable completions of quotient ordered semigroups

A poset is said to be ω-chain complete if every countable chain in it has a least upper bound. It is known that every partially ordered set has a natural ω-completion. In this paper we study the ω-completion of partially ordered semigroups, and the topological action of such a semigroup on its ω-completion. We show that, for partially ordered semigroups, ω-completion and quotient with respect to congruences are two operations that commute with each other. This contrasts with the case of general partially ordered sets.     

On countably compact,locally countable spaces

We study the cardinalities of countably compact, locally countableT ₃ spaces. For alln(<), there exists one of cardinality _n . IfV=L, then there exists one of cardinalityx iffx= orx =x. MA implies that there exists one of cardinality>2.     

On asymptotic dimension of countable Abelian groups

We compute the asymptotic dimension of the rationals given with an invariant proper metric. We also show that a countable torsion Abelian group taken with an invariant proper metric has asymptotic dimension zero.