Localy compact full homeomorphism groups are zero-dimensional
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- by James Keesling PDF
- Proc. Amer. Math. Soc. 29 (1971), 390-396 Request permission
Abstract:
Let X be a metric space and let $G(X)$ be the space of homeomorphisms of X with the compact open topology. If $G(X)$ is locally compact, then it is zero-dimensional. Some examples are also given of metric spaces X for which $G(X)$ is compact.References
-
R. D. Anderson, Spaces of homeomorphisms of finite graphs (preprint).
- Richard Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610. MR 19916, DOI 10.2307/2371787
- Anatole Beck, On invariant sets, Ann. of Math. (2) 67 (1958), 99–103. MR 92106, DOI 10.2307/1969929 N. Bourbaki, General topology. Part 2, Hermann, Paris; Addison-Wesley, Reading, Mass., 1966, Chapter 10. MR 34 #5044b. B. L. Brechner, Topological groups which are not full homeomorphism groups (preprint).
- Beverly L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516–548. MR 187208, DOI 10.1090/S0002-9947-1966-0187208-2
- H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. MR 220249, DOI 10.4064/fm-60-3-241-249
- Jean Dieudonné, On topological groups of homeomorphisms, Amer. J. Math. 70 (1948), 659–680. MR 26323, DOI 10.2307/2372204
- J. de Groot, Groups represented by homeomorphism groups, Math. Ann. 138 (1959), 80–102. MR 119193, DOI 10.1007/BF01369667
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- Robert Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc. 8 (1957), 372–373. MR 83681, DOI 10.1090/S0002-9939-1957-0083681-9
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104 J. Nagata, Modern dimension theory, Bibliotheca Mathematica, vol. 6, Interscience, New York, 1965. MR 34 #8380.
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 390-396
- MSC: Primary 54.80
- DOI: https://doi.org/10.1090/S0002-9939-1971-0275404-2
- MathSciNet review: 0275404