Title




Bornologies and metrically generated theories


Author






Abstract




Bornologies axiomatize an abstract notion of bounded sets and are introduced as collections of subsets satisfying a number of consistency properties. Bornological spaces form a topological construct, the morphisms of which are those functions which preserve bounded sets. A typical example is a bornology generated by a metric, i.e. the collection of all bounded sets for that metric. In a recent paper [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 15471556] the authors noted that many examples are known of natural functors describing the transition from categories of metric spaces to the metrizable objects in some given topological construct such that, in some natural way, the metrizable objects generate the whole construct. These constructs can be axiomatically described and are called metrically generated. The construct of bornological spaces is not metrically generated, but an important large subconstruct is. We also encounter other important examples of metrically generated constructs, the constructs of Lipschitz spaces, of uniform spaces and of completely regular spaces. In this paper, the unified setting of metrically generated theories is used to study the functorial relationship between these constructs and the one of bornological spaces. 


Language




English


Source (journal)




Topology and its applications.  Amsterdam


Publication




Amsterdam
:
2009


ISSN




01668641


DOI




10.1016/J.TOPOL.2008.12.018


Volume/pages




156
:7
(2009)
, p. 12241233


ISI




000264904500008


Full text (Publisher's DOI)






Full text (open access)





