the concept of an $l$-bornology is introduced and the theory of $l$-bornological spacesis being developed. in particular the lattice of all $l$-bornologies on a given set is studied and basic properties ofthe category of $l$-bornological spaces and bounded mappings are investigated.

The concept of an $L$-bornology is introduced and the theory of $L$-bornological spacesis being developed. In particular the lattice of all $L$-bornologies on a given set is studied and basic properties ofthe category of $L$-bornological spaces and bounded mappings are investigated.

An extension of bornological algebras with a bounded linear section gives rise to a six term exact sequence of the associated periodic and entire cyclic cohomology groups. 1. Introduction A convex vector bornology on a vector space is a collection of subsets satisfying some conditions 8]. A typical example is the collection of bounded subsets of a locally convex vector space. A bornological alg...

and Applied Analysis 3 that the function f : (X, α) → (Y, β) is strongly proximally continuous on A if ∀E ⊂ A, ∀S ⊂ X, EαS 󳨐⇒ f (E) βf (S) . (5) Finally, one says that f is strongly proximally continuous on B if f is strongly proximally continuous on A, for every A ∈ B. We shall use the notation C B (X, Y) to denote the family of the functions from X to Y which are strongly proximally continuou...

Motivated by Rosenthal’s famous $$l^1$$ -dichotomy in Banach spaces, Haydon’s theorem, and additionally recent works on tame dynamical systems, we introduce the class of locally convex spaces. This is a natural analogue Rosenthal spaces (for which any bounded sequence contains weak Cauchy subsequence). Our approach based bornology subsets turn closely related to eventual fragmentability. leads,...

The strong Whitney convergence on bornology introduced by Caserta in [9] is a generalization of the uniform Beer-Levi [5]. This paper aims to study some important topological properties space all real valued continuous functions metric endowed with topologies and bornology. More precisely, we investigate metrizability, various countability properties, countable tightness, Fr?chet property these...

In order to apply the concept of boundedness, so crucial in the theory of metric spaces, to the case of a general topological space Hu Sze-Tsen introduced the notions of a bornological space and of boundedness for a mapping of bornological spaces. In this work we discuss two alternative approaches how the concepts of bornology and boundedness can be extended to the case of fuzzy sets and many-v...