Pointwise bornological spaces

With each metric space (X,d) we can associate a bornological space (X,Bd) where Bd is the set of all subsets of X with finite diameter. Equivalently, Bd 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.