# Weak Topologies Determined by Distance Functionals

## Abstract

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.