Annales Mathématiques
Blaise Pascal

Let $K$ be a field with a Krull valuation $\left|\right|$ and value group $G\ne \left\{1\right\}$, and let $B_K$ be the valuation ring. Theories about spaces of countable type and Hilbert-like spaces in [1] and spaces of continuous linear operators in [2] require that all absolutely convex subsets of the base field $K$ should be countably generated as $B_K$-modules.

By [1] Prop. 1.4.1, the field $K$ is metrizable if and only if the value group $G$ has a cofinal sequence. We prove that for any fixed cardinality ${\aleph }_{\kappa }$, there exists a metrizable field $K$ whose value group has cardinality ${\aleph }_{\kappa }$. The existence of a cofinal sequence only depends on the choice of some appropriate ordinal $\alpha$ which has cardinality ${\aleph }_{\kappa }$ and which has cofinality $\omega$.

By [2] Prop. 1.4.4, the condition that any absolutely convex subset of $K$ be countably generated as a $B_K$-module is equivalent to the fact that the value group has a cofinal sequence and each element in the completion $G^{\#}$ is obtained as the supremum of a sequence of elements of $G$. We prove that for any fixed uncountable cardinal ${\aleph }_{\kappa }$ there exists a metrizable field $K$ of cardinality ${\aleph }_{\kappa }$ which has an absolutely convex subset that is not countably generated as a $B_K$-module.

We prove also that for any cardinality ${\aleph }_{\kappa }>{\aleph }_0$ for the value group the two conditions (the whole group has a cofinal sequence and every subset of the group which is bounded above has a cofinal sequence) are logically independent.

DOI : 10.5802/ambp.196

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     author = {Elena Olivos},
     title = {A family of totally ordered groups with some special properties},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {79--90},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {12},
     number = {1},
     year = {2005},
     doi = {10.5802/ambp.196},
     mrnumber = {2126442},
     zbl = {1085.06010},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.196/}
}

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Elena Olivos. A family of totally ordered groups with some special properties. Annales mathématiques Blaise Pascal, Tome 12 (2005) no. 1, pp. 79-90. doi : 10.5802/ambp.196. https://ambp.centre-mersenne.org/articles/10.5802/ambp.196/