Well-posedness, bornologies, and the structure of metric spaces

Gerald Beer, Manuel Segura

Abstract

Given a continuous nonnegative functional λ that makes sense defined on an arbitrary metric space (X, d), one may consider those spaces in which each sequence (xn) for which lim n→∞λ(xn) = 0 clusters. The compact metric spaces, the complete metric spaces, the cofinally complete metric spaces, and the UC-spaces all arise in this way. Starting with a general continuous nonnegative functional λ defined on (X, d), we study the bornology Bλ of all subsets A of X on which limn→∞λ(an) = 0 ⇒ (an) clusters, treating the possibility X ∈ Bλ as a special case. We characterize those bornologies that can be expressed as Bλ  for some λ, as well as those that can be so induced by a uniformly continuous λ.


Keywords

Well-posed problem; Bornology; UC-space; Cofinally complete space; Strong uniform continuity; Bornological convergence; Shielded from closed sets

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References

M.Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11-16.

http://dx.doi.org/10.2140/pjm.1958.8.11

H. Attouch, R. Lucchetti and R. Wets, The topology of the -Hausdorff distance, Ann. Mat. Pura Appl. 160 (1991), 303–320.

http://dx.doi.org/10.1007/BF01764131

H. Attouch and R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695–730.

J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Marcel Dekker, New York-Basel, 1980.

PMid:6986746

G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653–658.

http://dx.doi.org/10.1090/S0002-9939-1985-0810180-3

G. Beer, More about metric spaces on which continuous functions are uniformly continuous, Bull. Australian Math. Soc. 33 (1986), 397–406.

http://dx.doi.org/10.1017/S0004972700003981

G. Beer, UC spaces revisited. Amer. Math. Monthly 95 (1988), 737–739.

http://dx.doi.org/10.2307/2322255

G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, Holland, 1993.

G. Beer, On metric boundedness structures, Set-Valued Anal. 1 (1999), 195–208.

http://dx.doi.org/10.1023/A:1008720619545

G. Beer, Between compactness and completeness, Top. Appl. 155 (2008), 503–514.

http://dx.doi.org/10.1016/j.topol.2007.08.020

G. Beer, Operator topologies and graph convergence, J. Convex Anal., to appear.

G. Beer, C. Costantini and S. Levi, When is bornological convergence topological?, preprint. G. Beer and G. DiMaio, Cofinal completeness of the Hausdorff metric topology, preprint.

G. Beer and S. Levi, Pseudometrizable bornological convergence is Attouch-Wets convergence, J. Convex Anal. 15 (2008), 439–453.

G. Beer and S. Levi, Strong uniform continuity, J. Math. Anal. Appl. 350 (2009), 568–589.

http://dx.doi.org/10.1016/j.jmaa.2008.03.058

G. Beer, S. Naimpally and J. Rodríguez-López, S-topologies and bounded convergences, J. Math. Anal. Appl. 339 (2008), 542–552.

http://dx.doi.org/10.1016/j.jmaa.2007.07.010

J. Borwein and J. Vanderwerff, Epigraphical and uniform convergence of convex functions, Trans. Amer. Math. Soc. 348 (1996), 1617–1631.

http://dx.doi.org/10.1090/S0002-9947-96-01581-4

B. Burdick, On linear cofinal completeness, Top. Proc. 25 (2000), 435–455.

G. Di Maio, E. Meccariello, and S. Naimpally, Uniformizing (proximal) △-topologies, Top. Appl. 137 (2004), 99–113.

http://dx.doi.org/10.1016/S0166-8641(03)00202-5

A. Dontchev and T. Zolezzi, Well-posed optimization problems, Lecture Notes in Mathematics 143, Springer-Verlag, Berlin 1993.

J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

H. Hogbe-Nlend, Bornologies and functional analysis, North-Holland, Amsterdam, 1977.

A. Hohti, On uniform paracompactness, Ann. Acad. Sci. Fenn. Series A Math. Diss. 36 (1981), 1–46.

N. Howes, Modern analysis and topology, Springer, New York, 1995.

http://dx.doi.org/10.1007/978-1-4612-0833-4

S.-T. Hu, Boundedness in a topological space, J. Math Pures Appl. 228 (1949), 287–320.

S.-T. Hu, Intoduction to general topology, Holden-Day, San Francisco, 1966.

T. Jain and S. Kundu, Atsuji spaces: equivalent conditions, Topology Proc. 30 (2006), 301–325.

E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984.

K. Kuratowski, Topology vol. 1, Academic Press, New York, 1966.

A. Lechicki, S. Levi and A. Spakowski, Bornological convergences, J. Math. Anal. Appl. 297 (2004), 751–770.

http://dx.doi.org/10.1016/j.jmaa.2004.04.046

R. Lucchetti, Convexity and well-posed problems, Springer Verlag, Berlin, 2006.

S. Nadler and T. West, A note on Lesbesgue spaces, Topology Proc. 6 (1981), 363–369.

J.-P. Penot and C. Zalinescu, Bounded (Hausdorff) convergence : basic facts and applications, in Variational analysis and applications, F. Giannessi and A. Maugeri, eds., Kluwer Acad. Publ. Dordrecht, 2005.

http://dx.doi.org/10.1007/0-387-24276-7_49

J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math 9 (1959), 567–570.

http://dx.doi.org/10.2140/pjm.1959.9.567

M. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc. 62 (1977), 359–362.

http://dx.doi.org/10.1090/S0002-9939-1977-0436085-3

S. Romaguera, On cofinally complete metric spaces, Q & A in Gen. Top. 16 (1998), 165–170.

G. Toader, On a problem of Nagata, Mathematica (Cluj) 20 (1978), 77–79.

W. Waterhouse, On UC spaces, Amer. Math. Monthly 72 (1965), 634–635.

http://dx.doi.org/10.2307/2313854

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