Some results are obtained concerning n(T), the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if n(T) = 3 and T has a tight prime model, then T is unstable. Secondly, it is proved that if n(T) is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links...

An infinite first-order structure is minimal if its each definable subset is either finite or co-finite. We formulate three questions concerning order properties of minimal structures which are motivated by Pillay’s Conjecture (stating that a countable first-order structure must have infinitely many countable, pairwise non-isomorphic elementary extensions). In this article a connection between ...

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

Every complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomizat...

In this paper, spaces with σ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces with σ-locally countable cs-networks). To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia introduced the concept of weakly open mappings, by using it,...

Examples of spaces X are constructed for which Cp(X) is not normal but all closed discrete subsets are countable. A monolithic example is constructed in ZFC and a separable first countable example is constructed using ♦.

The Continuum Hypothesis implies an Erdös-Sierpiński like duality between the ideal of first category subsets of R , and the ideal of countable dimensional subsets of R. The algebraic sum of a Hurewicz subset a dimension theoretic analogue of Sierpinski sets and Lusin sets of R with any compactly countable dimensional subset of R has first category.

Extending work of [14, 16, 26], we show there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable L-spaces or compact S-spaces. The model is one of the form PFA(S)[S], where S is a coherent Souslin tree.

We define a compactum X to be AB-compact if the cofinality of the character χ(x, Y ) is countable whenever x ∈ Y and Y ⊂ X. It is a natural open question if every AB-compactum is necessarily first countable. We strengthen several results from [Arhangel’skii and Buzyakova, Convergence in compacta and linear Lindelöfness, Comment. Math. Univ. Carolin. 39 (1998), no. 1, 159–166] by proving the fol...

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal in [8] for the study of analytic equivalence relations, we study countable Bo...