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Keywords:
Hausdorff-Bourbaki quasi-uniformity; hyperspace; locally compact; cofinally complete; uniformly locally compact; co-uniformly locally compact
Summary:
We characterize those Tychonoff quasi-uniform spaces $(X,\mathcal {U})$ for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family $\mathcal {K}_{0}(X)$ of nonempty compact subsets of $X$. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space $X$ is uniformly locally compact on $\mathcal {K}_{0}(X)$ if and only if $X$ is paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is $\sigma $-compact if and only if its (lower) semicontinuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces $(X,\mathcal {U})$ for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on $\mathcal {K}_{0}(X)$ is obtained.
References:
[1] G.  Beer: Topologies on Closed and Convex Closed Sets. Mathematics and its Applications, Vol.  268. Kluwer Acad. Publ., , 1993. MR 1269778
[2] G.  Berthiaume: On quasi-uniformities in hyperspaces. Proc. Amer. Math. Soc. 66 (1977), 335–343. DOI 10.1090/S0002-9939-1977-0482620-9 | MR 0482620 | Zbl 0345.54026
[3] B. S.  Burdick: Local compactness of hyperspaces. Ann. New York Acad. Sci. 704 (1993), 28–33. DOI 10.1111/j.1749-6632.1993.tb52506.x | MR 1277840 | Zbl 0828.54021
[4] M. M.  Coban: Note sur la topologie exponentielle. Fund. Math. 71 (1971), 27–41. MR 0292014
[5] H. H.  Corson: The determination of paracompactness by uniformities. Amer. J.  Math. 80 (1958), 185–190. DOI 10.2307/2372828 | MR 0094780 | Zbl 0080.15803
[6] R.  Engelking: General Topology. Polish Sci. Publ., Warsaw, 1977. MR 0500780 | Zbl 0373.54002
[7] N. R.  Howes: Modern Analysis and Topology. University text. Springer-Verlag, New York, 1995. MR 1351251
[8] P.  Fletcher and W. F.  Lindgren: $C$-complete quasi-uniform spaces. Arch. Math. (Basel) 30 (1978), 175–180. DOI 10.1007/BF01226037 | MR 0474216
[9] P.  Fletcher and W. F.  Lindgren: Quasi-Uniform Spaces. Marcel Dekker, New York, 1982. MR 0660063
[10] H. P. A.  Künzi, M. Mršević, I. L.  Reilly and M. K.  Vamanamurthy: Convergence, precompactness and symmetry in quasi-uniform spaces. Math. Japonica 38 (1993), 239–253. MR 1213385
[11] H. P. A.  Künzi, S.  Romaguera: Left $K$-completeness of the Hausdorff quasi-uniformity. Rostock. Math. Kolloq. 51 (1997), 167–176.
[12] H. P. A.  Künzi, S.  Romaguera: Well-quasi-ordering and the Hausdorff quasi-uniformity. Topology Appl. 85 (1998), 207–218. DOI 10.1016/S0166-8641(97)00151-X | MR 1617464
[13] H. P. A.  Künzi and S.  Romaguera: Quasi-metric spaces, quasi-metric hyperspaces and uniform local compactness. Rend. Istit. Mat. Univ. Trieste 30 Suppl. (1999), 133–144. MR 1718967
[14] H. P. A.  Künzi and C.  Ryser: The Bourbaki quasi-uniformity. Topology Proc. 20 (1995), 161–183. MR 1429179
[15] E.  Michael: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152–182. DOI 10.1090/S0002-9947-1951-0042109-4 | MR 0042109 | Zbl 0043.37902
[16] I. L.  Reilly, P. V.  Subrahmanyam and M. K.  Vamanamurthy: Cauchy sequences in quasi-pseudo-metric spaces. Monatsh. Math. 93 (1982), 127–140. DOI 10.1007/BF01301400 | MR 0653103
[17] M. D.  Rice: A note on uniform paracompactness. Proc. Amer. Math. Soc. 62 (1977), 359–362. MR 0436085 | Zbl 0353.54011
[18] J.  Rodríguez-López and S.  Romaguera: The relationship between the Vietoris topology and the Hausdorff quasi-uniformity. Topology Appl 124 (2002), 451–464. DOI 10.1016/S0166-8641(01)00252-8 | MR 1930656
[19] S.  Romaguera: On hereditary precompactness and completeness in quasi-uniform spaces. Acta Math. Hungar. 73 (1996), 159–178. DOI 10.1007/BF00058951 | MR 1415928 | Zbl 0924.54035
[20] S.  Romaguera and M.  Sanchis: Locally compact topological groups and cofinal completeness. J.  London Math. Soc. 62 (2000), 451–460. DOI 10.1112/S0024610700001289 | MR 1783637
[21] M. A.  Sánchez-Granero: Covering axioms, directed GF-spaces and quasi-uniformities. Publ. Math. Debrecen 61 (2002), 357–381. MR 1943699
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