Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T01:45:39.605Z Has data issue: false hasContentIssue false
  • English
  • Français

On group uniformities on the square of a space and extending pseudometrics

Published online by Cambridge University Press:  17 April 2009

Michael G. Tkačnko
Michael G. Tkačnko
Affiliation:
Departamento de MatemáticasUniversidad Autonóma, MetropolitanaUnidad IztapalapaMexico 13 e-mail: mich@xanum.uam.mx
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.

We give some conditions under which, for a given pair (d1, d2) of continuous pseudometrics respectively on X and X3, there exists a continuous semi-norm N on the free topological group F(X) such that N(x · y−1) = d1(x, y) and N(x · y · t−1 · z−1) ≥ d2((x, y), (z, t)) for all x, y, z, t ∈ X. The “extension” results are applied to characterise thin subsets of free topological groups and obtain some relationships between natural uniformities on X2 and those induced by the group uniformities *V, V* and *V* of F(X).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 2 , April 1995 , pp. 309 - 335
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Comfort, W.W. and Ross, K.A., ‘Pseudocompactness and uniform continuity in topological groups’, Pacific J. Math. 16 (1966), 483–496.CrossRefGoogle Scholar
[2]Engelking, R., General topology (PWN, Warsaw, 1977).Google Scholar
[3]Graev, M.I., ‘Free topological groups’, (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 279–324. English translation: Transl. Math. Monographs 8 (1962), 305–364.Google Scholar
[4]Graev, M.I., ‘Theory of topological groups’, (in Russian), Uspekhi Mat. Nauk 5 (1950), 3–56.Google Scholar
[5]Markov, A.A., ‘On free topological groups’, (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945), 3–64. English translation: Transl. Math. Monographs 8 (1962), 195–272.Google Scholar
[6]Morris, S.A. and Nickolas, P., ‘Locally invariant topologies on free groups’, Pacific J. Math. 103 (1982), 523–537.Google Scholar
[7]Nummela, E.C., ‘Uniform free topological groups and Samuel compactifications’, Topology Appl. 13 (1982), 77–83.Google Scholar
[8]Pestov, V.G., ‘Some properties of free topological groups’, (in Russian), Vestnik Moscov. Univ. Ser. Matem. (1982), 35–37. English translation: Moscow Univ. Math. Bull. 37, 46–49.Google Scholar
[9]Pestov V.G., V.G., ‘Thin sets in topological groups and a new compactness theorem’, (in Russian), Izv. Vyssh. Uchebn. Zaved. Ser. Matem. 11 (1987), 64–66. English translation: Soviet Math. 31, 81–84.Google Scholar
[10]Sipacheva, O.V. and Tkačenko, M.G., ‘Thin and bounded subsets of free topological groups’, Topology Appl. 36 (1990), 143–156.Google Scholar
[11]Sipacheva, O.V. and Uspenskiiˇ, V.V., ‘Free topological groups with no small subgroups and Graev metrics’, (in Russian), Vestnik Moscov. Univ. Ser. Mat. 4 (1987), 21–24. English translation: Moscow Univ. Math. Bull. 42, 24–29.Google Scholar
[12]Šwierczkowski, S., ‘Topologies in free algebras’, Proc. London Math. Soc. 14 (1964), 566–576.Google Scholar
[13]Tkačenko, M.G., ‘On completeness of topological groups’, (in Russian), Sibirsk. Mat. J. 25 (1984), 146–158. English translation: Siberian Math. J. 25, 122–131.Google Scholar
[14]Tkačenko, M.G., ‘On topologies of free groups’, Czechoslovak. Math. J. 34 (1984), 541–551.CrossRefGoogle Scholar
[15]Tkačenko, M.G., ‘On some properties of free topological groups’, (in Russian), Mat. Zametki 37 (1985), 110–118. English translation: Math. Notes 37.Google Scholar
[16]Tkačenko, M.G., ‘Concordant extension of pseudometrics from subspaces of free topological groups’, in Algebraic and logic constructions (Collection), (in Russian) (Kalinin State Univ., 1989), pp. 94–103.Google Scholar