Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T04:16:55.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant metrics on free topological groups

Published online by Cambridge University Press:  17 April 2009

Sidney A. Morris  and
H.B. Thompson
Sidney A. Morris
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales;
H.B. Thompson
Affiliation:
School of Mathematical Sciences, Flinders University, Bedford Park, South Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 1 , August 1973 , pp. 83 - 88
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Abels, Herbert, “Normen auf freien topologischen Gruppen”, Math. Z. 129 (1972), 25–42.CrossRefGoogle Scholar
[2]Граев, М.И. [Graev, M.I.], “Свободные топологические группы” [Free topological groups], IzV. Akad. Nauk SSSR Ser. Mat. 12 (1948), 279–324; Amer. Math. Soc. Transl. 35 (1951); reprinted Amer. Math. Soc. Transl. (1) 8 (1962), 305–364.Google ScholarPubMed
[3]Grosser, Siegfried and Moskowitz, Martin, “Compactness conditions in topological groups”, J. reine angew. Math. 246 (1971), 1–40.Google Scholar
[4]Joiner, Charles, “Free topological groups and dimension”, (to appear)Google Scholar
[5]Kakutani, Shizuo, “Free topological groups and infinite direct product topological groups”, Proc. Imp. Acad. Tokyo 20 (1944), 595–598.Google Scholar
[6]Kelley, John L., General topology (Van Nostrand, Toronto, New York, London, 1955).Google Scholar
[7]Mack, John, Morris, Sidney A., Ordman, Edward T., “Free topological groups and the projective dimension of a locally compact abelian group”, Proc. Amer. Math. Soc. (to appear).Google Scholar
[8]Марков, А.А. [Markov, A.A.], “О свободных топологических группах” [On free topological groups], Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945), 3–64; Amer. Math. Soc. Transl. 30 (1950), 11–88; reprinted Amer. Math. Soc. Transl. (1) 8 (1962), 195–272.Google ScholarPubMed
[9]Michael, Ernest, “Bi-quotient maps and Cartesian products of quotient maps“, Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 2, 287–302, vii (1969).CrossRefGoogle Scholar
[10]Ordman, Edward T., “Free products of topological groups which are k w -spaces”, Trans. Amer. Math. Soc. (to appear).Google Scholar