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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The infimal value functional and the uniformization of hit-and-miss hyperspace topologies


Authors: Gerald Beer and Robert Tamaki
Journal: Proc. Amer. Math. Soc. 122 (1994), 601-612
MSC: Primary 54B20
MathSciNet review: 1264804
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Abstract: We give necessary and sufficient conditions for the uniformizability of hit-and-miss and proximal hit-and-miss hyperspace topologies defined on the nonempty closed subsets $ {\text{CL}}(X)$ of a Hausdorff uniform space $ \langle X,\mathcal{U}\rangle $. In the case of uniformizability, one can always find a family $ \mathcal{F}$ of continuous functions on X into [0, 1] so that the hyperspace topology is the weak topology induced by $ \{ {m_f}:f \in \mathcal{F}\} $, where for each f, $ {m_f}:{\text{CL}}(X) \to [0,1]$ is the infimal value functional defined by $ {m_f}(A) = \inf \{ f(x):x \in A\} $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1264804-5
PII: S 0002-9939(1994)1264804-5
Keywords: Hyperspace topology, hit-and-miss topology, uniformity, weak topology, infimal value functional, Vietoris topology, Fell topology
Article copyright: © Copyright 1994 American Mathematical Society