A note on bornologies
Journal Title: Математичні Студії - Year 2018, Vol 49, Issue 1
Abstract
A bornology on a set X is a family B of subsets of X closed under taking subsets, finite unions and such that ⋃B=X. We prove that, for a bornology B on X, the following statements are equivalent: (1) there exists a vector topology τ on the vector space V(X) over R such that B is the family of all subsets of X bounded in τ; (2) there exists a uniformity U on X such that B is the family of all subsets of X totally bounded in U; (3) for every Y⊆X, Y∉B, there exists a metric d on X such that B⊆Bd, Y∉Bd, where Bd is the family of all closed discrete subsets of (X,d); (4) for every Y⊆X, Y∉B, there exists Z⊆Y such that Z′∉B for each infinite subset Z′ of Z. A bornology B satisfying (4) is called antitall. We give topological and functional characterizations of antitall bornologies.
Authors and Affiliations
I. Protasov
Keywords
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- EP ID EP355307
- DOI 10.15330/ms.49.1.13-18
- Views 69
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