When Is a Family of Sets a Family of Bounded Sets?

Abstract

Let (X, r) be a metrizable topological space. A metric p for X is called admissible provided it is compatible with the topology r for X. Given any admissible metric p for the topology, the equivalent metric d defined by d(x, y) = min{1, p(x, y)} makes each subset of X d-bounded. Now let v be a family of subsets of X. In this article we address the following question: when does there exist an unbounded metric d compatible with r for which each member of v is a d-bounded set? Since points are always bounded with respect to any admissible metric, and bounded sets are closed under finite unions and are hereditary under inclusion, there is no loss in generality in assuming that (i) UV = X; (ii) {A1, A2,A3,.., An } c A n U>1A i E; (iii) A EV and B cA =B c V. A family of sets satisfying these properties is often called a bomology [1]. Thus, our question may be recast as follows: when can a bornology in a metrizable space be extended to a nontrivial metric bomology. i.e., a bornology consisting of the bounded subsets of X as determined by some unbounded admissible metric? We pause to introduce some notation and terminology. Let (X, r) be a metrizable space. We denote the interior, closure, and complement of a subset A of X by int A, cl A, and Ac, respectively. Given an admissible metric d for r, x c X, and r > 0 we denote the open (respectively, closed) ball of radius r and center x by S'(x) (respectively, B'(x)). A subset A of X is d-bounded if for some x c X and r > 0, A C Sd(x); equivalently, diamd(A) = sup{d(a1, a2): a1 cA, a2 c A} < oo. We denote the family of d-bounded sets by M(d), so that a bornology on X is a metric bornology if it coincides with W(d) for some admissible metric d. Finally, d(x, A) denotes inf {d(x, a): a c A}, the usual distance from a point x in X to a subset A of X. The question we posed at the outset has a rather simple answer. Recall that a family W of subsets of X is called discrete if each point x c X has a neighborhood V that meets at most one element of 9/.