Extending uniformly continuous pseudo-ultrametrics and uniform retracts
HTML articles powered by AMS MathViewer
- by Robert L. Ellis PDF
- Proc. Amer. Math. Soc. 30 (1971), 599-602 Request permission
Abstract:
It is first proved that any uniformly continuous pseudo-ultrametric on a subspace of a non-Archimedean uniform space X has a uniformly continuous extension to X (which preserves total boundedness or separability). Then it is proved that every complete subspace of an ultrametrizable space X is a uniform retract of X. This has consequences concerning the extension of uniformly continuous functions.References
- R. A. Alò and H. L. Shapiro, Extensions of totally bounded pseudometrics, Proc. Amer. Math. Soc. 19 (1968), 877–884. MR 232342, DOI 10.1090/S0002-9939-1968-0232342-9
- Richard Arens, Extension of coverings, of pseudometrics, and of linear-space-valued mappings, Canad. J. Math. 5 (1953), 211–215. MR 55666, DOI 10.4153/cjm-1953-023-1 N. Bourbaki, Elements of mathematics. General topology. Part I, Hermann, Paris; Addison-Wesley, Reading, Mass., 1966. MR 34 #5044a.
- Robert L. Ellis, Extending continuous functions on zero-dimensional spaces, Math. Ann. 186 (1970), 114–122. MR 261565, DOI 10.1007/BF01350686
- T. E. Gantner, Extensions of uniformly continuous pseudometrics, Trans. Amer. Math. Soc. 132 (1968), 147–157. MR 222836, DOI 10.1090/S0002-9947-1968-0222836-9
- J. R. Isbell, On finite-dimensional uniform spaces, Pacific J. Math. 9 (1959), 107–121. MR 105669
- A. F. Monna, Remarques sur les métriques non-archi-médiennes. I, Nederl. Akad. Wetensch., Proc. 53 (1950), 470–481 = Indagationes Math. 12, 122–133 (1950) (French). MR 35982
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 599-602
- MSC: Primary 54.30
- DOI: https://doi.org/10.1090/S0002-9939-1971-0283752-5
- MathSciNet review: 0283752