Open AccessRepresentations of bornologies
Author(s) -
Homeira Pajoohesh
Publication year - 2022
Publication title -
applied general topology
Language(s) - English
Resource type - Journals
eISSN - 1989-4147
pISSN - 1576-9402
DOI - 10.4995/agt.2022.16405
Subject(s) - mathematics , monoid , metric space , bounded function , equivalence relation , metric (unit) , equivalence (formal languages) , set (abstract data type) , combinatorics , euclidean space , discrete mathematics , power set , mathematical analysis , computer science , operations management , economics , programming language
Bornologies abstract the properties of bounded sets of a metric space. But there are unbounded bornologies on a metric space like $\mathcal{P}(\RR)$ with the Euclidean metric. We show that by replacing $[0,\infty)$ with a partially ordered monoid every bornology is the set of bounded subsets of a generalized metric mapped into a partially ordered monoid. We also prove that the set of bornologies on a set is the join completion of the equivalence classes of a relation on the power set of the set.