Abstract
A metric space is called cofinally complete if every cofinally Cauchy sequence in it has a cluster point. For a metric space (X, d), we consider the set AC(X) of almost nowhere locally compact sets (Beer and Di Maio in Acta Math Hung 134(3):322–343, 2012), which is a subset of the set of closed subsets in X. We consider some hyperspace topologies on the set AC(X) which it inherits from the set of non-empty closed subsets in X as subspace topologies and characterize cofinally complete metric spaces in terms of relations to Hausdorff metric topology, proximal topology, Vietoris topology and locally finite topology on AC(X). We characterize cofinally complete metric spaces in terms of some function space topologies as well. Furthermore, we characterize the class of metric spaces for which the corresponding space AC(X) equipped with the Hausdorff metric topology is cofinally complete.
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References
Aggarwal, M., Kundu, S.: More about the cofinally complete spaces and the Atsuji spaces. Houst. J. Math. 42(4), 1373–1395 (2016)
Atsuji, M.: Uniform continuity of continuous functions of metric spaces. Pac. J. Math. 8, 11–16 (1958)
Beer, G.: Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance. Proc. Am. Math. Soc. 95(4), 653–658 (1985)
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht (1993)
Beer, G.: Between compactness and completeness. Topol. Appl. 155, 503–514 (2008)
Beer, G., García-Lirola, L. C., Garrido, M. I.: Stability of Lipschitz-type functions under pointwise product and reciprocation. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(3), 16 (2020) (Paper No. 120)
Beer, G., Garrido, M.I.: Locally Lipschitz functions, cofinal completeness, and UC spaces. J. Math. Anal. Appl. 428(2), 804–816 (2015)
Beer, G., Garrido, M.I.: On the uniform approximation of Cauchy continuous functions. Topol. Appl. 208, 1–9 (2016)
Beer, G., Di Maio, G.: Cofinal completeness of the Hausdorff metric topology. Fund. Math. 208, 75–85 (2010)
Beer, G., Di Maio, G.: The Bornology of cofinally complete subsets. Acta Math. Hung. 134(3), 322–343 (2012)
Brandi, P., Ceppitelli, R., Holá, L.: Boundedly UC spaces and topologies on function spaces. Set-Valued Anal. 16, 357–373 (2008)
Corson, H.H.: The determination of paracompactness by uniformities. Am. J. Math. 80, 185–190 (1959)
García-Máynez, A., Romaguera, S.: Perfect pre-images of cofinally complete metric spaces. Comment. Math. Univ. Carolin. 40, 335–342 (1999)
Garrido, M.I., Jaramillo, J.A.: Lipschitz-type functions on metric spaces. J. Math. Anal. Appl. 340(1), 282–290 (2008)
Garrido, M.I., Meroño, Ana S.: New types of completeness in metric spaces. Ann. Acad. Sci. Fenn. Math. 39, 733–758 (2014)
Garrido, M.I., Meroño, Ana S.: On paracompactness, completeness and boundedness in uniform spaces. Topol. Appl 203, 98–107 (2016)
Gupta, L., Kundu, S.: Functions that preserve certain sequences and locally Lipschitz functions. Ann. Acad. Sci. Fenn. Math. 45, 699–722 (2020)
Gupta, L., Kundu, S.: Cofinal completion vis-á-vis Cauchy continuity and total boundedness. Topol. Appl. 290, 107576 (2020). https://doi.org/10.1016/j.topol.2020.107576
Holá, L’.: Hausdoff metric convergence of continuous functions. In: Proc. Sixth Prague Topological Symposium, pp. 263–271. Helderman Verlag, Berlin (1988)
Holá, L’.: Hausdorff metric on the space of upper semicontinuous multifunctions. Rocky Mt. J. Math. 22, 601–610 (1992)
Holá, L., Holý, D.: Further characterizations of boundedly UC spaces. Comment. Math. Univ. Carolin. 34, 175–183 (1993)
Holá, L’., Neubrunn, T.: On almost uniform convergence and convergence in Hausdorff metric. Rad. Mat. 4, 193–202 (1988)
Howes, N.R.: On completeness. Pac. J. Math. 38, 431–440 (1971)
Jain, T., Kundu, S.: Atsuji completions vis-à-vis hyperspaces. Math. Slovaca 58, 497–508 (2008)
Keremedis, K.: Metric spaces on which continuous functions are almost uniformly continuous. Topol. Appl. 232, 256–266 (2017)
Künzi, H.P.A., Romaguera, S.: Quasi-metric spaces, quasi-metric hyperspaces and uniform local compactness. Rend. Instit. Mat. Univ. Trieste. 30, 133–144 (1999)
Kundu, S., Jain, T.: Atsuji spaces: equivalent conditions. Topol. Proc. 30(1), 301–325 (2006)
Michael, E.: Topologies on spaces of subsets. Trans. Am. Math. Soc. 71, 152–182 (1951)
Naimpally, S.A.: Graph topology for function spaces. Trans. Am. Math. Soc. 123, 267–272 (1966)
Rice, M.D.: A note on uniform paracompactness. Proc. Am. Math. Soc. 62(2), 359–362 (1977)
Romaguera, S.: On cofinally complete metric spaces. Quest. Answ. Gen. Topol. 16(2), 165–170 (1998)
Sendov, Bl: Hausdorff Approximations. Kluwer Academic Publishers, Dordrecht (1990)
Smith, J.: Review of “A note on uniform paracompactness” by Michael D. Rice. Math. Rev. 55, (1978)
Tkachuk, V.V..: The space \(C_p(X)\) is cofinally Polish if and only if it is pseudocomplete. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(2), 10 (2021) (Paper No. 68)
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The authors would like to thank the Editor and the referees for their careful readings and valuable suggestions and for drawing the attention of the authors to [26].
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Gupta, L., Kundu, S. Cofinal completeness vis-á-vis hyperspaces. RACSAM 115, 82 (2021). https://doi.org/10.1007/s13398-021-01026-2
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DOI: https://doi.org/10.1007/s13398-021-01026-2
Keywords
- Hausdorff metric
- Continuous function
- Lipschitz-type functions
- Vietoris topology
- Locally finite topology
- Proximal topology
- Topology of uniform convergence
- Cofinally complete
- Almost nowhere locally compact