All hypertopologies are hit-and-miss

Somashekhar Naimpally

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.


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;

Full Text:

PDF

References

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Pub. (1993).

G. Beer, C. Himmelberg, C. Prikry and F. Van Vleck, The locally finite topology on 2X, Proc. Amer. Math. Soc. 101 (1987), 168-171.

G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. 162 (1992), 367-381. http://dx.doi.org/10.1007/BF01760016

G. Di Maio and S. Naimpally, Comparison of hypertopologies, Rend. Istit. Mat. Univ. Trieste 22 (1990), 140-161.

A. Di Concilio, S. Naimpally and P. Sharma, Proximal hypertopologies, Sixth Brazilian Topology Meeting, Campinas, Brazil (1988) (unpublished).

I. Del Prete and B. Lignola, On convergence of closed-valued multifunctions, Boll. Un. Mat. Ita. B 6 (1983), 819-834.

J. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472-476. http://dx.doi.org/10.1090/S0002-9939-1962-0139135-6

S. T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 28 (1949), 287-320.

J. Isbell, Uniform Spaces, American Mathematical Society (1964).

J. L. Kelly, General Topology, Van Nostrand (1955).

M. Marjanovic, Topologies on collections of closed subsets, Publ. Inst. Math. (Beograd) 20 (1966), 125-130.

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. http://dx.doi.org/10.1090/S0002-9947-1951-0042109-4

L. Nachman, Hyperspaces of proximity spaces, Math. Scand. 23 (1968), 201-213.

S. Naimpally and P. Sharma, Fine uniformity and the locally finite hyperspace topology, Proc. Amer. Math. Soc. 103 (1988), 641-646. http://dx.doi.org/10.1090/S0002-9939-1988-0943098-9

S. A. Naimpally and B. D.Warrack, Proximity Spaces, Cambridge Tracts in Mathematics 59, Cambridge University Press (1970).

H. Poppe, Eine Bemerkung über Trennungsaxiome im Raum der abgeschlossenen Teilmengen eines topologischen Raumes, Arch. Math. 16 (1965), 197-199. http://dx.doi.org/10.1007/BF01220021

D. H. Smith, Hyperspaces of a uniformizable spaces, Proc. Camb. Phil. Soc. 62 (1966), 25-28. http://dx.doi.org/10.1017/S0305004100039487

A. J. Ward, A counter-example in uniformity theory, Proc. Camb. Phil. Soc. 62 (1966), 207-208. http://dx.doi.org/10.1017/S0305004100039761

A. J. Ward, On H-equivalence of uniformties: the Isbell-Smith problem, Pacific J. Math. 22 (1967), 189-196. http://dx.doi.org/10.2140/pjm.1967.22.189

F. Wattenberg, Topologies on the set of closed subsets, Pacific J. Math. 68 (1977), 537-551. http://dx.doi.org/10.2140/pjm.1977.68.537

R. Wijsman, Convergence of sequences of convex sets, cones and functions, II, Trans. Amer. Math. Soc. 123 (1966), 32-45. http://dx.doi.org/10.1090/S0002-9947-1966-0196599-8

Abstract Views

1334
Metrics Loading ...

Metrics powered by PLOS ALM


 

Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. A short note on hit-and-miss hyperspaces
René Bartsch, Harry Poppe
Applied General Topology  vol: 4  issue: 2  first page: 281  year: 2003  
doi: 10.4995/agt.2003.2033



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt