Bulletin of the Belgian Mathematical Society - Simon Stevin

Metrizability of totally ordered groups of infinite rank and their completions

A. Mansilla, E. Olivos, and H. Soto

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In [4], Ochsenius and Schikhof ask the following question. Given a totally ordered group $G$ with a cofinal sequence, if every element of its Dedekind completion $G^\#$ is the supremum of a sequence in $G$, does it follow that $G^\#$ is metrizable? We answer their question by studying topological properties of a family of totally ordered groups, $\Gamma_\alpha$, and their completions $\Gamma_\alpha^\#$. Furthermore we obtain for this family conditions both necessary and sufficient for the metrizability of $\Gamma_\alpha^\#$.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 14, Number 5 (2007), 969-977.

First available in Project Euclid: 17 December 2007

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Zentralblatt MATH identifier

Primary: 06F30: Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12] 54E35: Metric spaces, metrizability
Secondary: 06F15: Ordered groups [See also 20F60] 22B99: None of the above, but in this section

Metrizability Topological ordered groups Topological G-modules


Olivos, E.; Soto, H.; Mansilla, A. Metrizability of totally ordered groups of infinite rank and their completions. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 5, 969--977. doi:10.36045/bbms/1197908907. https://projecteuclid.org/euclid.bbms/1197908907

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