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Beer, Gerald; Tamaki, Robert

The infimal value functional and the uniformization of hit-and-miss hyperspace topologies. (English) Zbl 0824.54003

Proc. Am. Math. Soc. 122, No. 2, 601-612 (1994).
Summary: We give necessary and sufficient conditions for the uniformizability of hit-and-miss and proximal hit-and-miss hyperspace topologies defined on the nonempty closed subsets \(\text{CL} (X)\) of a Hausdorff uniform space \((X,{\mathcal U})\). In the case of uniformizability, one can always find a family \({\mathcal F}\) of continuous functions on \(X\) into \([0,1]\) so that the hyperspace topology is the weak topology induced by \(\{m_ f : f \in {\mathcal F}\}\), where for each \(f\), \(m_ f : \text{CL} (X) \to [0,1]\) is the infimal value functional defined by \(m_ f (A) = \inf \{f(x) : x \in A\}\).

Cited in 9 Documents

MSC:

54B20 Hyperspaces in general topology

Keywords:

Fell topology; Vietoris topology; proximal hit-and-miss hyperspace topologies; infimal value functional
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\textit{G. Beer} and \textit{R. Tamaki}, Proc. Am. Math. Soc. 122, No. 2, 601--612 (1994; Zbl 0824.54003)
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References:

[1] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0561.49012
[2] Hédy Attouch and Gerald Beer, On the convergence of subdifferentials of convex functions, Arch. Math. (Basel) 60 (1993), no. 4, 389 – 400. · Zbl 0778.49018
[3] Gerald Beer, Support and distance functionals for convex sets, Numer. Funct. Anal. Optim. 10 (1989), no. 1-2, 15 – 36. · Zbl 0696.46010
[4] Gerald Beer, The slice topology: a viable alternative to Mosco convergence in nonreflexive spaces, Nonlinear Anal. 19 (1992), no. 3, 271 – 290. · Zbl 0786.46006
[5] Gerald Beer, Lipschitz regularization and the convergence of convex functions, Numer. Funct. Anal. Optim. 15 (1994), no. 1-2, 31 – 46. · Zbl 0817.49018
[6] Gerald Beer and Jonathan M. Borwein, Mosco and slice convergence of level sets and graphs of linear functionals, J. Math. Anal. Appl. 175 (1993), no. 1, 53 – 67. · Zbl 0781.46016
[7] Gerald Beer, Alojzy Lechicki, Sandro Levi, and Somashekhar Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. (4) 162 (1992), 367 – 381. · Zbl 0774.54004
[8] Gerald Beer and Roberto Lucchetti, Weak topologies for the closed subsets of a metrizable space, Trans. Amer. Math. Soc. 335 (1993), no. 2, 805 – 822. · Zbl 0810.54011
[9] Gerald Beer and Robert K. Tamaki, On hit-and-miss hyperspace topologies, Comment. Math. Univ. Carolin. 34 (1993), no. 4, 717 – 728. · Zbl 0787.54013
[10] G. Choquet, Outils topologiques et métriques de l’analyse mathématique, Centre de documentation universitaire et S.E.D.E.S. réunis, Paris, 1966.
[11] A. Di Concilio, S. Naimpally, and P. Sharma, Proximal hypertopologies, preprint.
[12] Giuseppe Di Maio and Ľubica Holá, On hit-and-miss topologies, Rend. Accad. Sci. Fis. Mat. Napoli (4) 62 (1995), 103 – 124 (1996) (English, with English and Italian summaries). · Zbl 1162.54308
[13] V. A. Efremovič, The geometry of proximity. I, Mat. Sbornik N. S. 31(73) (1952), 189 – 200 (Russian).
[14] J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472 – 476. · Zbl 0106.15801
[15] Sebastiano Francaviglia, Alojzy Lechicki, and Sandro Levi, Quasiuniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), no. 2, 347 – 370. · Zbl 0587.54003
[16] C. Hess, Contributions à l’ètude de la mesurabilité, de la loi de probabilité, et de la convergence des multifunctions. Thèse d’état, U.S.T.L. Montpellier, 1986.
[17] J.-L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. (9) 52 (1973), 421 – 441 (1974) (French). · Zbl 0282.46005
[18] Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. · Zbl 0556.28012
[19] Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152 – 182. · Zbl 0043.37902
[20] S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. · Zbl 0206.24601
[21] Harry Poppe, Eine Bemerkung über Trennungsaxiome in Räumen von abgeschlossenen Teilmengen topologischer Räume, Arch. Math. (Basel) 16 (1965), 197 – 199 (German). · Zbl 0127.13004
[22] Harry Poppe, Einige Bemerkungen über den Raum der abgeschlossenen Mengen, Fund. Math. 59 (1966), 159 – 169 (German). · Zbl 0139.40404
[23] Yves Sonntag and Constantin Zălinescu, Set convergences. An attempt of classification, Trans. Amer. Math. Soc. 340 (1993), no. 1, 199 – 226. · Zbl 0786.54013
[24] Wim Vervaat, Random upper semicontinuous functions and extremal processes, Probability and lattices, CWI Tract, vol. 110, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1997, pp. 1 – 56. · Zbl 0882.60003
[25] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32 – 45. · Zbl 0146.18204
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