Bombay hypertopologies

Authors

  • Giuseppe Di Maio Seconda Università degli Studi di Napoli
  • Enrico Meccariello Università del Sannio
  • Somashekhar Naimpally

DOI:

https://doi.org/10.4995/agt.2003.2042

Keywords:

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 ∆-topology

Abstract

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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Author Biographies

Giuseppe Di Maio, Seconda Università degli Studi di Napoli

Dipartimento di Matematica

Enrico Meccariello, Università del Sannio

Facoltà di Ingegneria

References

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

G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Annali di Matematica pura ed applicata 162 (1992), 367-381. http://dx.doi.org/10.1007/BF01760016

C. Costantini, S. Levi and J. Zieminska, Metrics that generate the same hyperspace convergence, Set-Valued Analysis 1 (1993), 141-157. http://dx.doi.org/10.1007/BF01027689

D. Di Caprio and E. Meccariello, Notes on Separation Axioms in Hyperspaces, Q. & A. in General Topology 18 (2000), 65-86.

D. Di Caprio and E. Meccariello, G-uniformities, LR-proximities and hypertopologies, Acta Math. Hungarica 88 (1-2) (2000), 73-93. http://dx.doi.org/10.1023/A:1006752510935

A. Di Concilio, S. Naimpally and P.L. Sharma, Proximal hypertopologies, Proceedings of the VI Brasilian Topological Meeting, Campinas, Brazil (1988) [unpublished].

G. Di Maio and L. Holá, A hypertopology determined by the family of totally bounded sets is the infimum of upper Wijsman topologies, Q. & A. in General Topology, 15 (1997), 51-66.

G. Di Maio and D. Holý, Comparison among Wijsman topology and other hypertopologies, Atti Sem. Mat. Fis. Univ. di Modena 48 (2000), 121-133.

G. Di Maio, E. Meccariello and S. Naimpally, Uniformizing (proximal) ∆-topologies, Topology and its Applications, (to appear).

G. Di Maio and S. Naimpally, Comparison of hypertopologies, Rendiconti di Trieste 22 (1990), 140-161.

G. Di Maio and S. Naimpally, Abstract measure of farness and Wijsmann convergence, Zbornik Radova 5 (1991), 109-112.

G. Di Maio and S. Naimpally, Some notes on hyperspace topologies, Ricerche di Matematica, (to appear).

R. Engelking, General topology, Revised and completed version, Helderman Verlag, (Helderman, Berlin, 1989.)

S. Francaviglia, A. Lechicki and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370. http://dx.doi.org/10.1016/0022-247X(85)90246-X

L. Holá and R. Lucchetti, Equivalence among hypertopologies, Set-Valued Analysis 3 (1995), 339-350. http://dx.doi.org/10.1007/BF01026245

S.T. Hu, Boundedness in topological space, J. Math. Pures. Appl. 28 (1949), 287-340.

M. Marjanovic, Topologies on collections of closed subsets, Publ. Inst. Math. (Beograd) 20 (1966), 196-130.

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

C.J. Mozzochi, M. Gagrat and S. Naimpally, Symmetric generalized topological structures, Exposition Press, (Hicksville, New York, 1976.)

S. Naimpally, All Hypertopologies are hit-and-miss, Applied General Topology 3 (2002), 45-53.

S. Naimpally and P. Sharma, Fine uniformity and locally finite hyperspace topology on 2X, Proc. Amer. Math. Soc. 103 (1988), 641-646. http://dx.doi.org/10.1090/S0002-9939-1988-0943098-9

S. Naimpally, B. Warrack, Proximity spaces, Cambridge Tracts in Mathematics 59, (Cambridge University Press, 1970.)

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

W.J. Thron, Proximity Structures and Grills, Math. Ann. 206 (1973), 35-62. http://dx.doi.org/10.1007/BF01431527

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

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Published

2003-10-01

How to Cite

[1]
G. Di Maio, E. Meccariello, and S. Naimpally, “Bombay hypertopologies”, Appl. Gen. Topol., vol. 4, no. 2, pp. 421–444, Oct. 2003.

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