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...

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...

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.

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...

Salas and Tapia-García introduced the concept of an extended locally convex space in [13] which extends idea normed (introduced by Beer [2]). This article gives attractive formulation finest topology provides a systematic study resulting space. As application, we characterize coincidence topologies corresponding to uniform strong convergences on bornology for function C(X).

It is shown that every locally idempotent (locallym-pseudoconvex) Hausdorff algebra A with pseudoconvex vonNeumannbornology is a regular (respectively, bornological) inductive limit of metrizable locallym-(kB-convex) subalgebras AB of A. In the case where A, in addition, is sequentially BA-complete (sequentially advertibly complete), then every subalgebraAB is a locally m-(kB-convex) Fréchet al...