It is well-known that on quasi-pseudometric space $(X,q)$, every $q^s$-Cauchy sequence left (or right) $K$-Cauchy but the converse does not hold in general. In this article, we study a class of maps preserve (right) sequences call sequentially-regular maps. Moreover, characterize totally bounded sets terms and right uniformly locally semi-Lipschitz

The complexity (quasi-pseudo-metric) spaces have been introduced as part of the development of a topological foundation for the complexity analysis of algorithms ((Sch95]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in Sch95]. Typically these applications involve xed point arguments based on the Smyth completion. The notion of S-com...

The Kočinac αi properties, i = 1, 2, 3, 4, are generalizations of Arhangel’skĭı’s αi local properties. We give a complete classification of these properties when applied to the standard families of open covers of topological spaces or to the standard families of open covers of topological groups. One of the latter properties characterizes totally bounded groups. We also answer a question of Koč...

The notion of completeness on L-semi-uniform spaces is introduced in this paper. We have established that completeness is weakly hereditary and preserved under L-semi-uniform isomorphism. Finally, we have shown that in a totally bounded L-semi-uniform space, completeness is equivalent to compactness. 2010 AMS Classification: 54A40

Let F = F(t, x) be a bounded, Hausdorff continuous multifunction with compact, totally disconnected values. Given any y0 ∈ F(t0, x0), we show that the differential inclusion ẋ ∈ F(t, x)⊂ Rm has a globally defined classical solution, with x(t0)= x0, ẋ(t0)= y0. © 2008 Elsevier Inc. All rights reserved.

Modifying Bowen’s entropy, we introduce a new uniform entropy. We prove that the completion theorem for uniform entropy holds in the class of all metric spaces. However, the completion theorem for Bowen’s entropy does not hold in the class of all totally bounded metric spaces.

We show a natural graph-theoretic generalization of the Sauer–Shelah lemma. This result is applied to bound the ‘∞ and L1 packing numbers of classes of functions whose range is an arbitrary, totally bounded metric space. ? 1998 Elsevier Science B.V. All rights reserved.

In the present paper we prove the a.e. convergence of Fejér means of integrable functions with respect to the two-dimensional representative product systems on a bounded compact totally disconnected group provided that the set of indices is in a cone-like set.

We show that and how the Coulomb potential V (x) = Z e/x can be regularized and solved exactly at the imaginary coupling Z e. The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case.

We study the infimum of the Hausdorff and Vietoris topologies on the hyperspace of a metric space. We show that this topology coincides with the supremum of the upper Hausdorff and lower Vietoris topologies if and only if the underlying metric space is either totally bounded or is a UC space.