Bombay Hypertopologies


G. Di Maio, E. Meccariello, S. Naimpally

Recently it was shown that, in a metric space, the topology corresponding to the Wijsman convergence, is a new hit-and-miss hypertopology called the far-miss topology. This topology is a mixture of the ball topology t(B) and the proximal ball topology s(B), where B denotes the ring generated by all proper closed balls of nonnegative radii and the proximity involved is the metric one. In this paper, we introduce a new hypertopology, which we call the Bombay topology. This hypertopology includes as special cases, all the finite hypertopologies including the above three. This facilitates the comparison of all finite hypertopologies. We point out some errors in the literature. It is possible to get similar results for locally finite hypertopologies, including the newly born locally finite Wijsman, locally finite ball and locally finite proximal ball topologies.


Key words and phrases: 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, D-topology, proximal locally finite D-topology.


A.M.S. 2000 Classification: 54B20, 54A10, 54C35, 54D30, 54E05, 54E15.


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