Bombay hypertopologies
DOI:
https://doi.org/10.4995/agt.2003.2042Keywords:
Hyperspace, Wijsman topology, Ball topology, Proximal ball topology, Far-miss topology, hit-and-miss topology, Bombay topology, Vietoris topology, Fell topology, Proximal locally finite topology, ∆-topology, Proximal locally finite ∆-topologyAbstract
Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.
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