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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
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Abstract

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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

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