A generalization of boundedly compact metric spaces
Beer, Gerald ; Di Concilio, Anna
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991), p. 361-367 / Harvested from Czech Digital Mathematics Library

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.

Publié le : 1991-01-01
Classification:  54B20,  54C35,  54E15,  54E45
@article{116977,
     author = {Gerald Beer and Anna Di Concilio},
     title = {A generalization of boundedly compact metric spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {32},
     year = {1991},
     pages = {361-367},
     zbl = {0766.54028},
     mrnumber = {1137797},
     language = {en},
     url = {http://dml.mathdoc.fr/item/116977}
}
Beer, Gerald; Di Concilio, Anna. A generalization of boundedly compact metric spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 361-367. http://gdmltest.u-ga.fr/item/116977/

Atsuji M. Uniform continuity of continuous functions on metric spaces, Pacific J. Math. 8 (1958), 11-16. (1958) | MR 0099023

Atsuji M. Uniform continuity of continuous functions on uniform spaces, Pacific J. Math. 13 (1961), 657-663. (1961) | MR 0165489 | Zbl 0102.37703

Attouch H.; Wets R. Quantitative stability of variational systems: I. The epigraphical distance, to appear, Trans. Amer. Math. Soc. | MR 1018570 | Zbl 0753.49007

Attouch H.; Lucchetti R.; Wets R. The topology of the $\rho $-Hausdorff distance, to appear, Annali Mat. Pura Appl. | Zbl 0769.54009

Azé D.; Penot J.-P. Operations on convergent families of sets and functions, Optimization 21 (1990), 521-534. (1990) | MR 1069660

Beer G. Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653-658. (1985) | MR 0810180 | Zbl 0594.54007

Beer G. More about metric spaces on which continuous functions are uniformly continuous, Bull. Australian Math. Soc. 33 (1986), 397-406. (1986) | MR 0837486 | Zbl 0573.54026

Beer G. UC spaces revisited, Amer. Math. Monthly 95 (1988), 737-739. (1988) | MR 0966244 | Zbl 0656.54022

Beer G. Convergence of continuous linear functionals and their level sets, Archiv der Math. 52 (1989), 482-491. (1989) | MR 0998621 | Zbl 0662.46015

Beer G. Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117-126. (1990) | MR 0982400 | Zbl 0681.46014

Beer G.; Di Concilio A. Uniform continuity on bounded sets and the Attouch-Wets topology, to appear, Proc. Amer. Math. Soc. | MR 1033956 | Zbl 0677.54007

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) | MR 0897090

Beer G.; Lucchetti A. Convex optimization and the epi-distance topology, to appear, Trans. Amer. Math. Soc. | MR 1012526 | Zbl 0681.46013

Beer G.; Lucchetti A. Weak topologies for the closed subsets of a metrizable space, preprint. | Zbl 0810.54011

Castaing C.; Valadier M. Convex analysis and measurable multifunctions, Lecture Notes in Mathematics No. 580, Springer-Verlag, Berlin, 1977. | MR 0467310 | Zbl 0346.46038

Di Concilio A.; Naimpally S. Atsuji spaces - continuity versus uniform continuity, in Proc. VI Brazilian Conf. on Topology, Campinǫs-Sao Paulo, August 1988.

Hausdorff H. Erweiterung einer Homöomorphie, Fund. Math. 16 (1930), 353-360. (1930)

Holá L. The Attouch-Wets topology and a characterization of normable linear spaces, to appear, Bull. Australian Math. Soc. | MR 1120389

Hueber H. On uniform continuity and compactness in metric spaces, Amer. Math. Monthly 88 (1981), 204-205. (1981) | MR 0619571 | Zbl 0451.54024

Levine N. Remarks on uniform continuity in metric spaces, Amer. Math. Monthly 67 (1979), 562-563. (1979) | MR 0116310

Michael E. Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. (1951) | MR 0042109 | Zbl 0043.37902

Monteiro A.; Peixoto M. Le nombre de Lebesgue et la continuité uniforme, Portugaliae Math. 10 (1951), 105-113. (1951) | MR 0044608 | Zbl 0045.25801

Nagata J. On the uniform topology of bicompactifications, J. Inst. Polytech. Osaka City University 1 (1950), 28-38. (1950) | MR 0037501 | Zbl 0041.51601

Penot J.-P. The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity, to appear, Proc. Amer. Math. Soc. | MR 1068129 | Zbl 0774.54008

Rainwater J. Spaces whose finest uniformity is metric, Pacific J. Math. 9 (1959), 567-570. (1959) | MR 0106448 | Zbl 0088.38301

Revalski J.; Zhivkov N. Well-posed optimization problems in metric spaces, preprint.

Sendov Bl. Hausdorff approximations, Bulgarian Academy of Sciences, Sofia, 1979 (in Russian); English version published by Kluwer, Dordrecht, Holland, 1990. | MR 1078632 | Zbl 0715.41001

Toader Gh. On a problem of Nagata, Mathematica (Cluj) 20 (43) (1978), 78-79. (1978) | MR 0530953 | Zbl 0409.54041

Vaughan H. On locally compact metrizable spaces, Bull. Amer. Math. Soc. 43 (1937), 532-535. (1937) | MR 1563581

Waterhouse W. On UC spaces, Amer. Math. Monthly 72 (1965), 634-635. (1965) | MR 0184200 | Zbl 0136.19802