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.

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,...

In this paper, we introduce the notation of soft bornological group to solve problem boundedness for group. We combine set theory with bornology space produce a new structure which is called So that both product and inverse maps are bounded. As well as, study actions on sets. The aim partition into orbital classes by acting set. addition, explain centralizer, normalizer, stabilizer in details. ...

We primarily make a general approach to the study of open covers and related selection principles using idea statistical convergence in metric space. In process we are able extend some results (Caserta et al. 2012; Chandra 2020) where bornological spaces have been investigated strong uniform (Beer Levi, 2009) on bornology. introduce notion statistical-Bs-cover, statistically-strong-B-Hurewicz s...

The notion of locally quasi-convex abelian group, introduce by Vilenkin, is extended to maximally almost-periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set rep(G) of all finite dimensional continuous representations on a topological group G in order to associate well behaved group topologies (dual topologies) to them. As a co...

A topological space X is called A-real compact, if every algebra homomorphism from A to the reals is an evaluation at some point of X, where A is an algebra of continuous functions. Our main interest lies on algebras of smooth functions. In [AdR] it was shown that any separable Banach space is smoothly real compact. Here we generalize this result to a huge class of locally convex spaces includi...