On Chains in <Emphasis Type="BoldItalic">H</Emphasis>-Closed Topological Pospaces

We study chains in an H-closed topological partially ordered space. We give sufficient conditions for a maximal chain L in an H-closed topological partially ordered space (H-closed topological semilattice) under which L contains a maximal (minimal) element. We also give sufficient conditions for a linearly ordered topological partially ordered space to be H-closed. We prove that a linearly ordered H-closed topological semilattice is an H-closed topological pospace and show that in general, this is not true. We construct an example of an H-closed topological pospace with a non-H-closed maximal chain and give sufficient conditions under which a maximal chain of an H-closed topological pospace is an H-closed topological pospace.     

On linearly ordered <Emphasis Type="Italic">H</Emphasis>-closed topological semilattices

We give a criterion for a linearly ordered topological semilattice to be H-closed. We also prove that any linearly ordered H-closed topological semilattice is absolutely H-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an H-closed topological semilattice.     

On generalized <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> and <Emphasis Type="Italic">R</Emphasis><Subscript>1</Subscript> spaces

Characterizations of some properties of generalized R ₀ and R ₁ topological spaces by using closure operator defined on a generalized topological space will be given. It is also shown that many results done in this area in some previous papers can be considered as special cases of our results.     

Topological <Emphasis Type="Italic">K</Emphasis>-equivalence of analytic function-germs

We study the topological K-equivalence of function-germs (ℝ ⁿ , 0) → (ℝ, 0). We present some special classes of piece-wise linear functions and prove that they are normal forms for equivalence classes with respect to topological K-equivalence for definable functions-germs. For the case n = 2 we present polynomial models for analytic function-germs.     

Combinatorial independence in measurable dynamics

We develop a fine-scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IE-tuples and measure IN-tuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations of the Pinsker von Neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence, ?₁ geometry, and Voiculescu's completely positive approximation entropy. Among the novel features of our local study is the treatment of general discrete acting groups, with the structural assumption of amenability in the case of entropy.     

Hierarchies in φ‐spaces and applications

We establish some results on the Borel and difference hierarchies in φ‐spaces. Such spaces are the topological counterpart of the algebraic directed‐complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non‐collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question of characterizing the ω‐ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological properties of the multifunction space <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) of cusco maps

A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology. The second author is supported by the SPM fellowship awarded by the Council of Scientific and Industrial Research, India.     

Independence entropy of \mathbb{Z}^{d}-shift spaces

We introduce a concept of independence entropy for symbolic dynamical systems. This notion of entropy measures the extent to which one can freely insert symbols in positions without violating the constraint defined by the shift space. We show that for a certain class of one-dimensional shift spaces X, the independence entropy coincides with the limiting, as d tends to infinity, topological entropy of the dimensional shift defined by imposing the constraints of X in each of the d cardinal directions. This is of interest because for these shift spaces independence entropy is easy to compute. Thus, while in these cases, the topological entropy of the d-dimensional shift (d≥2) is difficult to compute, the limiting topological entropy is easy to compute. In some cases, we also compute the rate of convergence of the sequence of d-dimensional entropies. This work generalizes earlier work on constrained systems with unconstrained positions.     

The <Emphasis Type="Italic">C</Emphasis>-topology on lattice-ordered groups

Let A be a lattice-ordered group. Gusi′c showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi′c's theorem,and reveal the very nature of a "C-group" of Gusi′c in this paper. Moreover,we show that the C-topological groups are topological lattice-ordered groups,and prove that every archimedean lattice-ordered vector space is a T2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T2....     

On C‐Suslin spaces

We prove a closed graph theorem for Baire locally convex spaces (for Baire linear topological spaces) in the domain and weakly C‐Suslin locally convex spaces (respectively, for C‐Suslin linear topological spaces) in the range which improves some classic closed graph theorems and other, more recent, related results.     

Neighbourhoods of independence and associated geometry in manifolds of bivariate Gaussian and Freund distributions

We provide explicit information geometric tubular neighbourhoods containing all bivariate distributions sufficiently close to the cases of independent Poisson or Gaussian processes. This is achieved via affine immersions of the 4-manifold of Freund bivariate distributions and of the 5-manifold of bivariate Gaussians. We provide also the α-geometry for both manifolds. The Central Limit Theorem makes our neighbourhoods of independence limiting cases for a wide range of bivariate distributions; the topological character of the results makes them stable under small perturbations, which is important for applications in models of stochastic processes.      

Intuitionistic <Emphasis Type="Italic">I</Emphasis>-fuzzy topological spaces

The main purpose of this paper is to introduce the concept of intuitionistic I-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points. The relation between the category of intuitionistic I-fuzzy topological spaces and the category of intuitionistic I-fuzzy quasi-coincident neighborhood spaces are studied. By using fuzzifying topology, the notion of generated intuitionistic I-fuzzy topology is proposed, and the connections among generated intuitionistic I-fuzzy topological spaces, fuzzifying topological spaces and I-fuzzy topological spaces are discussed. Finally, the properties of the operators Iω, ι are obtained.     

<Emphasis Type="BoldItalic">C</Emphasis>(<Emphasis Type="BoldItalic">X</Emphasis>)-objects in the Category of Semi-affine Lattices

In our previous paper (Hušek and Pulgarín, Topol Appl, doi:, 2009) we characterized the set C(X) of real-valued continuous functions on a topological space X as a real ℓ-group. The present paper weakens the situation to the level of semi-affine lattices.     

On a characterization theorem for locally compact abelian groups

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.     

On fuzzy semi δ-<Emphasis Type="Italic">V</Emphasis> continuity in fuzzy δ-<Emphasis Type="Italic">V</Emphasis> topological space

New concepts of fuzzy semi δ-V and fuzzy semi δ-Λ sets were introduced in our work “On fuzzy semi δ-Λ sets and fuzzy semi δ-V sets V-6,” J. Trip. Math. Soc., 6, 81–88 (2004). It was shown that the family of all fuzzy semi δ-V sets forms a fuzzy supra topological space on X denoted by (X, FS ^δV ). The aim of this paper is to introduce the concept of fuzzy semi δ-V continuity in a fuzzy δ-V topological space. Finally, some properties, preservation theorems, etc., are studied. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 712–717, May, 2008.     

Linear Independence of the Values of <Emphasis Type="Italic">q</Emphasis>-Hypergeometric Series and Related Functions

We give linear independence results for the values of certain entire series and of functions satisfying certain first order q-difference equations. The former generalizes a result of Bézivin, while the latter refines that of the second named author in qualitative form. These results imply linear independence of the values of q-hypergeometric series.Research supported in part by Grant-in-Aid for Scientic Research (No. 13640007), the Ministry of Education, Science, Sports and Culture of Japan.2000 Mathematics Subject Classification: Primary—11J72     

Some remarks on <Emphasis Type="Italic">S</Emphasis>-paracompact spaces

Some characterizations of S-paracompact spaces are given. We introduce a class of S-expandable spaces and study topological properties of S-expandable spaces. The second author acknowledges the support from the NSFC (No. 10571081) and the Natural Science Foundation of Jingsu Higher Education Institutions of China (Grant No. 07KJB110055).     

An application of Newton–Puiseux charts to the Jacobian problem

We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps (f,g) whose topological degree is ≤5, for maps with gcd(degf,degg)≤16 and for maps with. gcd(degf,degg) equal to 2 times a prime.