All hypertopologies are hit-and-miss

Authors

  • Somashekhar Naimpally Lakehead University

DOI:

https://doi.org/10.4995/agt.2002.2111

Keywords:

Hypertopology, Vietoris topology, Hausdorff metric, Hausdorff-Bourbaki uniformity, Uniformity, Proximal topology, Hit-and-miss topology, Locally finite, Wijsman topology, Proximal ball topology, Ball topology, Far-miss topology, Bounded Vietoris topology,

Abstract

We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in”nice” metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-and-miss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T1 space. Several existing results in the literature are easy consequences of our work.

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Author Biography

Somashekhar Naimpally, Lakehead University

Prof. Emeritus of Mathematics

Lakehead University

96 Dewson Street

Toronto, Ontario

M6H 1H3 CANADA

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Published

2002-04-01

How to Cite

[1]
S. Naimpally, “All hypertopologies are hit-and-miss”, Appl. Gen. Topol., vol. 3, no. 1, pp. 45–53, Apr. 2002.

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Articles