Convex fuzzifying bornologies and its duality

Abstract

The main objective of this paper is to investigate convex fuzzifying
bornological linear spaces and their duality. First, we introduce
the notions of convex fuzzifying bornological linear spaces and
locally convex fuzzifying topological linear spaces, along with
several examples. Next, we study the relationship between convex
fuzzifying bornological linear spaces and locally convex fuzzifying
topological linear spaces, utilizing the mapping of fuzzifying
bornivorous. Subsequently, we introduce the concepts of {\it
fuzzifying bornological locally convex spaces} and {\it fuzzifying
topological convex bornological linear spaces}. We demonstrate that
the category of {\it fuzzifying bornological locally convex spaces}
can be embedded as a reflective subcategory in the category of
convex fuzzifying bornological linear spaces, and the category of
{\it fuzzifying topological convex bornological linear spaces} can
also be embedded as a reflective subcategory in the the category
locally convex fuzzifying topological linear spaces. Moreover, the
category of {\it fuzzifying topological convex bornological linear
spaces} is topological over the category of linear spaces with
respect to the expected forgetful functor. Lastly, we provide
several characterizations of the fuzzifying bornological
topologies.