Weak Topologies Determined by Distance Functionals

  • Gerald Beer
Part of the Mathematics and Its Applications book series (MAIA, volume 268)


Let <X,d> be a metric space. In classical analysis, we think of the distance d(x,A) from a point xX 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 Ad(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.

Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Gerald Beer
    • 1
  1. 1.Department of Mathematics and Computer ScienceCalifornia State UniversityLos AngelesUSA

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