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Ordered products of topological groups

Published online by Cambridge University Press:  24 October 2008

M. Henriksen ,
R. Kopperman  and
F. A. Smith
M. Henriksen
Affiliation:
Harvey Mudd College, Claremont, CA 91711, U.S.A.
R. Kopperman
Affiliation:
City College of New York, New York, NY 10031, U.S.A.
F. A. Smith
Affiliation:
Kent State University, Kent, OH 44242, U.S.A.
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Extract

The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 281 - 295
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Bigard, A., Keimel, K. and Wolfenstein, S.. Groupes et Anneaux Réticulés, Lecture Notes in Math. vol. 608 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[2]Eilenberg, S.. Ordered topological spaces. Amer. J. Math. 63 (1941), 39–45.CrossRefGoogle Scholar
[3]Fuchs, L.. Partially Ordered Algebraic Systems (Pergamon Press, 1963).Google Scholar
[4]Hall, M.. The Theory of Groups (Macmillan, 1959).Google Scholar
[5]Hewitt, E. and Koshi, S.. Orderings in locally compact groups and the theorems of F. and M. Riesz, Math. Proc. Cambridge Philos. Soc. 93 (1983), 441–487.CrossRefGoogle Scholar
[6]Kelley, J.. General Topology (Van Nostrand, 1955).Google Scholar
[7]Kokorin, A. and Kopytov, V.. Fully Ordered Groups (John Wiley and Sons, 1974).Google Scholar
[8]Mura, R. and Rhemtulla, A.. Orderable Groups. Lecture Notes in Pure and Appl. Math. 27 (Marcel Dekker, 1977).Google Scholar