Orderability and continuous selections for Wijsman and Vietoris hyperspaces

Debora Di Caprio, Stephen Watson

Abstract

Bertacchi and Costantini obtained some conditions equivalent to the existence of continuous selections for the Wijsman hyperspace of ultrametric Polish spaces. We introduce a new class of hypertopologies, the macro-topologies. Both the Wijsman topology and the Vietoris topology belong to this class. We show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. In the setting of Polish spaces, these conditions are substantially weaker than the ones given by Bertacchi and Costantini. In particular, we conclude that Polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the Wijsman topology, just as it is for [0; 1]. This also solves a problem implicitely raised in Bertacchi and Costantini's paper.


Keywords

Selection; Vietoris topology; Wijsman topology; macro-topology; ∆-topology; Ordered space; Compatible order; Sub-compatible order; Extra-dense set; Lexor; Complete lexor; Polish space; Star-set; n-coordinated-function; n-coordinated-set

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References

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

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

D. Bertacchi, C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology and its Applications 88 (1998), 179-197. http://dx.doi.org/10.1016/S0166-8641(97)00175-2

E. Cech, Topological Spaces (Wiley, New York, 1966).

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

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

G. Di Maio, L. Holá, On hit-and-miss topologies, Rend. Acc. Sc. fis. mat. Napoli LXII (1995), 103-124.

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

R. Engelking, R.V. Heath, E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150-158. http://dx.doi.org/10.1007/BF01425452

W. Fleischman (Ed.), Set-valued mappings, selections, and topological properties of 2X, Lecture Notes in Mathematics 171 (Springer-Verlag, New York, 1970).

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

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

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

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Universitat Politècnica de València

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