Geodesic
A generalization of boundedly compact metric spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 361-367 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 54B20, 54C35, 54E15, 54E45
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 
@article{CMUC_1991_32_2_a17,
     author = {Beer, Gerald and Di Concilio, Anna},
     title = {A generalization of boundedly compact metric spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {361--367},
     year = {1991},
     volume = {32},
     number = {2},
     mrnumber = {1137797},
     zbl = {0766.54028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a17/}
}
TY  - JOUR
AU  - Beer, Gerald
AU  - Di Concilio, Anna
TI  - A generalization of boundedly compact metric spaces
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 1991
SP  - 361
EP  - 367
VL  - 32
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a17/
LA  - en
ID  - CMUC_1991_32_2_a17
ER  - 
%0 Journal Article
%A Beer, Gerald
%A Di Concilio, Anna
%T A generalization of boundedly compact metric spaces
%J Commentationes Mathematicae Universitatis Carolinae
%D 1991
%P 361-367
%V 32
%N 2
%U http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a17/
%G en
%F CMUC_1991_32_2_a17
Beer, Gerald; Di Concilio, Anna. A generalization of boundedly compact metric spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 361-367. http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a17/

[At1] Atsuji M.: Uniform continuity of continuous functions on metric spaces. Pacific J. Math. 8 (1958), 11-16. | MR

[At2] Atsuji M.: Uniform continuity of continuous functions on uniform spaces. Pacific J. Math. 13 (1961), 657-663. | MR | Zbl

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

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

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

[Be1] Beer G.: Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance. Proc. Amer. Math. Soc. 95 (1985), 653-658. | MR | Zbl

[Be2] Beer G.: More about metric spaces on which continuous functions are uniformly continuous. Bull. Australian Math. Soc. 33 (1986), 397-406. | MR | Zbl

[Be3] Beer G.: UC spaces revisited. Amer. Math. Monthly 95 (1988), 737-739. | MR | Zbl

[Be4] Beer G.: Convergence of continuous linear functionals and their level sets. Archiv der Math. 52 (1989), 482-491. | MR | Zbl

[Be5] Beer G.: Conjugate convex functions and the epi-distance topology. Proc. Amer. Math. Soc. 108 (1990), 117-126. | MR | Zbl

[BDC] Beer G., Di Concilio A.: Uniform continuity on bounded sets and the Attouch-Wets topology. to appear, Proc. Amer. Math. Soc. | MR | Zbl

[BHPV] Beer G., Himmelberg C., Prikry K., Van Vleck F.: The locally finite topology on $2^X$. Proc. Amer. Math. Soc. 101 (1987), 168-172. | MR

[BL1] Beer G., Lucchetti A.: Convex optimization and the epi-distance topology. to appear, Trans. Amer. Math. Soc. | MR | Zbl

[BL2] Beer G., Lucchetti A.: Weak topologies for the closed subsets of a metrizable space. preprint. | Zbl

[CV] Castaing C., Valadier M.: Convex analysis and measurable multifunctions. Lecture Notes in Mathematics No. 580, Springer-Verlag, Berlin, 1977. | MR | Zbl

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

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

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

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

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

[Mi] Michael E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152-182. | MR | Zbl

[MP] Monteiro A., Peixoto M.: Le nombre de Lebesgue et la continuité uniforme. Portugaliae Math. 10 (1951), 105-113. | MR | Zbl

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

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

[Ra] Rainwater J.: Spaces whose finest uniformity is metric. Pacific J. Math. 9 (1959), 567-570. | MR | Zbl

[RZ] Revalski J., Zhivkov N.: Well-posed optimization problems in metric spaces. preprint.

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

[To] Toader Gh.: On a problem of Nagata. Mathematica (Cluj) 20 (43) (1978), 78-79. | MR | Zbl

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

[Wa] Waterhouse W.: On UC spaces. Amer. Math. Monthly 72 (1965), 634-635. | MR | Zbl