Symmetric Bombay topology
Submitted: 2013-11-12
|Accepted: 2013-11-12
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Abstract:
The subject of hyperspace topologies on closed or closed and compact subsets of a topological space X began in the early part of the last century with the discoveries of Hausdorff metric and Vietoris hit-and-miss topology. In course of time, several hyperspace topologies were discovered either for solving some problems in Applied or Pure Mathematics or as natural generalizations of the existing ones. Each hyperspace topology can be split into a lower and an upper part. In the upper part the original set inclusion of Vietoris was generalized to proximal set inclusion. Then the topologization of the Wijsman topology led to the upper Bombay topology which involves two proximities. In all these developments the lower topology, involving intersection of finitely many open sets, was generalized to locally finite families but intersection was left unchanged. Recently the authors studied symmetric proximal topology in which proximity was used for the first time in the lower part replacing intersection with its generalization: nearness. In this paper we use two proximities also in the lower part and we obtain the lower Bombay hypertopology. Consequently, a new hypertopology arises in a natural way: the symmetric Bombay topology which is the join of a lower and an upper Bombay topology.
References:
G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, 1993.
M. Coban, Note sur la topologie exponentielle, Fund. Math. LXXI (1971), 27–41.
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P.L. Sharma, Proximal hypertopologies, Proceedings of the VI Brasilian Topological Meeting, Campinas, Brazil (1988) [unpublished].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, On hit-and miss topologies, Rend. Acc. Sc. Fis. Mat. Napoli 57 (1995), 103–124.
G. Di Maio, E. Meccariello and S. Naimpally, Bombay hypertopologies, Applied General Topology 4 (2) (2003), 421–424.
G. Di Maio, E. Meccariello and S. Naimpally, Symmetric proximal hypertopology, Rostock. Math. Kolloq. 58 (2003), 2–25.
G. Di Maio, E. Meccariello and S. Naimpally, Uniformizing (proximal) -topologies, Topology and its Applications, 137(2004), 99–113. http://dx.doi.org/10.1016/S0166-8641(03)00202-5
R. Engelking, General topology, Revised and completed version, Helderman Verlag, Helderman, Berlin, 1989.
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
F. Hausdorff, Grundz¨uge der Mengenlehre, Leipzing, 1914.
D. Harris, Regular-closed spaces and proximities, Pacif. J. Math. 34 (1970), 675–685. http://dx.doi.org/10.2140/pjm.1970.34.675
D. Harris, Completely regular proximities and RC-proximities, Fund. Math. LXXXV (1974), 103–111.
L. Holá and S. Levi, Decomposition properties of hyperspace topologies, Set-Valued Analysis 5 (1997), 309–321. http://dx.doi.org/10.1023/A:1008608209952
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, A short hystory of hyperspace topologies, in preparation.
S. Naimpally and B. Warrack, Proximity spaces, Cambridge Tracts in Mathematics 59, Cambty Press, 1970.
H. Poppe, Eine Bemerkungüber Trennungsaumen abgeschlossenen Teilmengen topologischer Räume, Arch. Math. 16 (1965), 197–199. http://dx.doi.org/10.1007/BF01220021
H. Poppe, Einigee Bemerkungen ¨uber den R¨schlossenen Mengen, Fund. Math. 59 (1966), 159–169.
R. F. Snipes, Functions that preserve Cauchy sequences, Nieuw Arch. Voor Wiskunde (3) 25 (1977), 409–422.
W. J. Thron, Topological Structures, Holt, Rinehart and Winston, New York, 1966.
L. Vietoris, Bereiche zweiter Ordnung, Monatsh. fur Math. und Phys. 32 (1922), 258–280. http://dx.doi.org/10.1007/BF01696886
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
S. Willard, General Topology, Addison-Wesley, 1970.
