Uniform convergence for a hyperspace
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- by R. J. Gazik PDF
- Proc. Amer. Math. Soc. 42 (1974), 302-306 Request permission
Abstract:
In this note a uniform convergence in the collection $C(E)$ of nonempty, compact subsets of a separated uniform convergence space E is defined. This convergence is compared with the hyperspace convergence on $C(E)$ and it is shown that the two convergences agree on Richardson’s class $\Gamma$. In the case of a regular ${T_1}$ topological space (E, t) this means that there is a uniform convergence structure on E, which induces t, such that uniform convergence in $C(E)$ is convergences in the Vietoris topology on $C(E)$.References
-
A. C. Cochran, On uniform convergence structures and convergence spaces, Doctoral Dissertation, University of Oklahoma, Norman, Okla., 1966.
- C. H. Cook and H. R. Fischer, Uniform convergence structures, Math. Ann. 173 (1967), 290–306. MR 217756, DOI 10.1007/BF01781969
- H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269–303 (German). MR 109339, DOI 10.1007/BF01360965
- R. J. Gazik, A hyperspace for convergence spaces, Proc. Amer. Math. Soc. 37 (1973), 234–240. MR 309042, DOI 10.1090/S0002-9939-1973-0309042-1
- H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334–341 (German, with English summary). MR 225280, DOI 10.1007/BF02052894
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- G. D. Richardson, A class of uniform convergence structures, Proc. Amer. Math. Soc. 25 (1970), 399–402. MR 256335, DOI 10.1090/S0002-9939-1970-0256335-X
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 302-306
- MSC: Primary 54B20
- DOI: https://doi.org/10.1090/S0002-9939-1974-0331304-3
- MathSciNet review: 0331304