• The conditions are investigated under which the upper Vietoris topology coincides with Scott on K ( X ) . For a well-filtered space having first-countable Smyth power space, and coincide. T 0 in set of minimal elements any compact saturated subset, its is first-countable. Alexandroff double circle, first-countable, not , let be poset all non-empty sets reverse inclusion order. said to have pr...

Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 subsets of 2 , under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of Pw. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of PM .

We de*ne the clique-width of a countable graph. We prove that a countable graph has *nite clique-width i+ its *nite induced subgraphs have bounded clique-width. We obtain an application to a conjecture concerning the structure of sets of countable graphs having a decidable monadic second-order satis*ability problem. c © 2003 Elsevier B.V. All rights reserved.

We consider random processes more general than those considered by Erdös and Rényi for generating the countable random graph. It is proved that, in the category sense, almost all random processes we consider generate the countable random graph with probability 1. Under a weak boundedness assumption we give a criterion for the random processes which generate the countable random graph almost sur...

Abstract In this paper, we establish a connection between categorical closedness and nontopologizability of semigroups. particular, for the class