Weak Topologies Determined by Distance Functionals
Let <X,d> be a metric space. In classical analysis, we think of the distance d(x,A) from a point x ∈ X to a nonempty closed subset A of X as a function of the point argument x with the set argument A held fixed. However, we may equally well regard this assignment as a function of the set argument with the point held fixed. We write d(x, ·) for the assignment A → d(x,A) on the nonempty closed subsets CL(X) of X. In this chapter, we consider weak topologies on CL(X) determined by families of distance functionals. A particular family of distance functionals is determined by the range of two parameters: the point x in X and the metric d. Ordinarily, the point x is allowed to run freely over the underlying space X. On the other hand, the parameter d may represent a fixed metric, or one chosen from a particular class of metrics.