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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Beer, G. (1993). Weak Topologies Determined by Distance Functionals. In: Topologies on Closed and Closed Convex Sets. Mathematics and Its Applications, vol 268. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8149-3_2
Download citation
DOI: https://doi.org/10.1007/978-94-015-8149-3_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4333-7
Online ISBN: 978-94-015-8149-3
eBook Packages: Springer Book Archive