A family of totally ordered groups with some special properties
Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 1, p. 79-90

Let K be a field with a Krull valuation || and value group G{1}, 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 κ , there exists a metrizable field K whose value group has cardinality κ . The existence of a cofinal sequence only depends on the choice of some appropriate ordinal α which has cardinality κ and which has cofinality ω.

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 κ there exists a metrizable field K of cardinality κ which has an absolutely convex subset that is not countably generated as a B K -module.

We prove also that for any cardinality κ > 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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