A generalization of boundedly compact metric spaces

Gerald Beer; Anna Di Concilio

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 2, page 361-367
  • ISSN: 0010-2628

Abstract

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A metric space X , d is called a UC space provided each continuous function on X into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that UC spaces play relative to the compact metric spaces.

How to cite

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Beer, Gerald, and Di Concilio, Anna. "A generalization of boundedly compact metric spaces." Commentationes Mathematicae Universitatis Carolinae 32.2 (1991): 361-367. <http://eudml.org/doc/247275>.

@article{Beer1991,
abstract = {A metric space $\langle X,d\rangle $ is called a $\operatorname\{UC\}$ space provided each continuous function on $X$ into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that $\operatorname\{UC\}$ spaces play relative to the compact metric spaces.},
author = {Beer, Gerald, Di Concilio, Anna},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\operatorname\{UC\}$ space; boundedly $\operatorname\{UC\}$ space; boundedly compact space; Atsuji space; uniform continuity on bounded sets; topology of uniform convergence on bounded sets; Attouch–Wets topology; UC spaces; boundedly compact spaces; set of nonisolated points; Hausdorff metric; Attouch-Wets topology},
language = {eng},
number = {2},
pages = {361-367},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A generalization of boundedly compact metric spaces},
url = {http://eudml.org/doc/247275},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Beer, Gerald
AU - Di Concilio, Anna
TI - A generalization of boundedly compact metric spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 2
SP - 361
EP - 367
AB - A metric space $\langle X,d\rangle $ is called a $\operatorname{UC}$ space provided each continuous function on $X$ into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that $\operatorname{UC}$ spaces play relative to the compact metric spaces.
LA - eng
KW - $\operatorname{UC}$ space; boundedly $\operatorname{UC}$ space; boundedly compact space; Atsuji space; uniform continuity on bounded sets; topology of uniform convergence on bounded sets; Attouch–Wets topology; UC spaces; boundedly compact spaces; set of nonisolated points; Hausdorff metric; Attouch-Wets topology
UR - http://eudml.org/doc/247275
ER -

References

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  1. Atsuji M., Uniform continuity of continuous functions on metric spaces, Pacific J. Math. 8 (1958), 11-16. (1958) MR0099023
  2. Atsuji M., Uniform continuity of continuous functions on uniform spaces, Pacific J. Math. 13 (1961), 657-663. (1961) Zbl0102.37703MR0165489
  3. Attouch H., Wets R., Quantitative stability of variational systems: I. The epigraphical distance, to appear, Trans. Amer. Math. Soc. Zbl0753.49007MR1018570
  4. Attouch H., Lucchetti R., Wets R., The topology of the ρ -Hausdorff distance, to appear, Annali Mat. Pura Appl. Zbl0769.54009
  5. Azé D., Penot J.-P., Operations on convergent families of sets and functions, Optimization 21 (1990), 521-534. (1990) MR1069660
  6. Beer G., Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653-658. (1985) Zbl0594.54007MR0810180
  7. Beer G., More about metric spaces on which continuous functions are uniformly continuous, Bull. Australian Math. Soc. 33 (1986), 397-406. (1986) Zbl0573.54026MR0837486
  8. Beer G., UC spaces revisited, Amer. Math. Monthly 95 (1988), 737-739. (1988) Zbl0656.54022MR0966244
  9. Beer G., Convergence of continuous linear functionals and their level sets, Archiv der Math. 52 (1989), 482-491. (1989) Zbl0662.46015MR0998621
  10. Beer G., Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117-126. (1990) Zbl0681.46014MR0982400
  11. Beer G., Di Concilio A., Uniform continuity on bounded sets and the Attouch-Wets topology, to appear, Proc. Amer. Math. Soc. Zbl0677.54007MR1033956
  12. Beer G., Himmelberg C., Prikry K., Van Vleck F., The locally finite topology on 2 X , Proc. Amer. Math. Soc. 101 (1987), 168-172. (1987) MR0897090
  13. Beer G., Lucchetti A., Convex optimization and the epi-distance topology, to appear, Trans. Amer. Math. Soc. Zbl0681.46013MR1012526
  14. Beer G., Lucchetti A., Weak topologies for the closed subsets of a metrizable space, preprint. Zbl0810.54011
  15. Castaing C., Valadier M., Convex analysis and measurable multifunctions, Lecture Notes in Mathematics No. 580, Springer-Verlag, Berlin, 1977. Zbl0346.46038MR0467310
  16. Di Concilio A., Naimpally S., Atsuji spaces - continuity versus uniform continuity, in Proc. VI Brazilian Conf. on Topology, Campinǫs-Sao Paulo, August 1988. 
  17. Hausdorff H., Erweiterung einer Homöomorphie, Fund. Math. 16 (1930), 353-360. (1930) 
  18. Holá L., The Attouch-Wets topology and a characterization of normable linear spaces, to appear, Bull. Australian Math. Soc. MR1120389
  19. Hueber H., On uniform continuity and compactness in metric spaces, Amer. Math. Monthly 88 (1981), 204-205. (1981) Zbl0451.54024MR0619571
  20. Levine N., Remarks on uniform continuity in metric spaces, Amer. Math. Monthly 67 (1979), 562-563. (1979) MR0116310
  21. Michael E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. (1951) Zbl0043.37902MR0042109
  22. Monteiro A., Peixoto M., Le nombre de Lebesgue et la continuité uniforme, Portugaliae Math. 10 (1951), 105-113. (1951) Zbl0045.25801MR0044608
  23. Nagata J., On the uniform topology of bicompactifications, J. Inst. Polytech. Osaka City University 1 (1950), 28-38. (1950) Zbl0041.51601MR0037501
  24. Penot J.-P., The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity, to appear, Proc. Amer. Math. Soc. Zbl0774.54008MR1068129
  25. Rainwater J., Spaces whose finest uniformity is metric, Pacific J. Math. 9 (1959), 567-570. (1959) Zbl0088.38301MR0106448
  26. Revalski J., Zhivkov N., Well-posed optimization problems in metric spaces, preprint. 
  27. Sendov Bl., Hausdorff approximations, Bulgarian Academy of Sciences, Sofia, 1979 (in Russian); English version published by Kluwer, Dordrecht, Holland, 1990. Zbl0715.41001MR1078632
  28. Toader Gh., On a problem of Nagata, Mathematica (Cluj) 20 (43) (1978), 78-79. (1978) Zbl0409.54041MR0530953
  29. Vaughan H., On locally compact metrizable spaces, Bull. Amer. Math. Soc. 43 (1937), 532-535. (1937) MR1563581
  30. Waterhouse W., On UC spaces, Amer. Math. Monthly 72 (1965), 634-635. (1965) Zbl0136.19802MR0184200

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