Let $K$ be a field with a Krull valuation $\left|\phantom{\rule{0.277778em}{0ex}}\right|$ 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.

@article{AMBP_2005__12_1_79_0, author = {Olivos, Elena}, title = {A family of totally ordered groups with some special properties}, journal = {Annales math\'ematiques Blaise Pascal}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {12}, number = {1}, year = {2005}, pages = {79-90}, doi = {10.5802/ambp.196}, mrnumber = {2126442}, zbl = {1085.06010}, language = {en}, url = {http://www.numdam.org/item/AMBP_2005__12_1_79_0} }

Olivos, Elena. A family of totally ordered groups with some special properties. Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 1, pp. 79-90. doi : 10.5802/ambp.196. http://www.numdam.org/item/AMBP_2005__12_1_79_0/

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