Pointwise bornological spaces

With each metric space ( X , d ) we can associate a bornological space ( X , B d ) where B d is the set of all subsets of X with finite diameter. Equivalently, B d is the set of all subsets of X that are contained in a ball with finite radius. If the metric d can attain the value infinite, then the set of all subsets with finite diameter is no longer a bornology. Moreover, if d is no longer symmetric, then the set of subsets with finite diameter does not coincide with the set of subsets that are contained in a ball with finite radius. In this text we will introduce two structures that capture the concept of boundedness in both symmetric and non-symmetric extended metric spaces.