Excess Topologies in Metric Spaces


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New Zealand Journal of Mathematics

Vol. 44, (2014), Pages 45-60

Gerald Beer

Department of Mathematics,

California State University Los Angeles,

5151 State University Drive, Los Angeles,

California 90032, USA


Camillo Costantini

Dipartimento di Matematica,

Università di Torino,

via Carlo Alberto 10,

10123 Torino, Italy


Sandro Levi

Dipartimento di Matematica e Applicazioni,

Università di Milano-Bicocca,

via Cozzi 53,

20125 Milano, Italy


Abstract Given a family \mathcal {S} of nonempty subsets of a metric space \langle X,d \rangle containing the singletons, we consider topologies on the nonempty subsets of X generated by families of excess functionals of the from \{e_d(S,\cdot) : S \in \mathcal{S}\}. Such topologies can be broken into lower and upper halves: the lower (resp. upper) half is the weakest topology on the nonempty subsets such that each excess functional in the family is upper (resp. lower) semicontinuous. Remarkably, one such topology can be stronger than another while its lower half can fail to be. We also study excess topologies generated by families of the form \{e_d(\cdot,S) : S \in \mathcal{S}\}; the results we give exhibit a decided lack of symmetry with those for the former class of excess topologies. This paper can be viewed as a sequel to a recent paper by the authors on gap topologies [3]. While the methodology and point-of-view are similar, the subject matter here is considerably more subtle.

Keywords excess functional, excess topology, Wijsman topology, dual Wijsman topology, Hausdorff distance, gap functional, quasi-uniformity, strict inclusion, bornology

Classification (MSC2000) 54B20.

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