Definition of 'completeness' of a valued field

Question

I have on a number of occasions seen reference to 'complete valued fields' without any explanation of precisely what completeness means in this context. It is definitely not referring to maximal completeness or spherical completeness, but rather something along the lines of Cauchy completeness. Although most of the literature refers to complete discrete valued fields, some of it appears to indicate that completeness also works in a non-discrete setting.

Most importantly, I need to understand the notion of completeness referred to in Proposition 6 of this paper, where it is claimed that completeness follows from maximal completeness. I add the definition of a valued field that I am working with below to avoid confusion:

A valued field is a field $K$ equipped with a valuation, which is a surjective map $v:K\to G\cup\lbrace\infty\rbrace$ where $v(x)=\infty\Leftrightarrow x=0$ and $$v(xy)=v(x)+v(y),$$ $$v(x+y)\geq\min\lbrace v(x),v(y)\rbrace$$ for all $x,y\in K$ and where $G$ is a totally ordered abelian group, called its value group.

Is there a common understanding of what 'completeness' means in the context of valued fields?

Answer 1

They are definitely referring to Cauchy completeness, i.e. the assertion that every Cauchy net converges (you could also use filters if you prefer; note it is important to use nets/filters in lieu of sequences because the space may not be first countable). For $g\in G$ one can define the subset $U_g:=\{x\in K\mid v(x)>g\}$, and then there is a unique topology on $K$ for which $\{U_g\mid g\in G\}$ is a neighborhood basis of $0$, and this topology makes $K$ into a topological field. For any topological abelian group the following notions make sense (note: you probably don't actually need the word "abelian" here):

Definition: If $A$ is a topological abelian group, then a net $(x_i)_{i\in I}$ is Cauchy if for every neighborhood $U$ of $0$, there exists some $i_0\in I$ such that $x_i-x_j\in U$ for all $i,j\ge i_0$.

Definition: A net $(x_i)_{i\in I}$ converges to $x\in A$ if for every neighborhood $U$ of $x$ there exists some $i_0\in I$ such that $x_i\in U$ for all $i\ge i_0$.

Definition: A topological abelian group is complete if every Cauchy net converges.

Just in case you want full assurance this is what the author means, here is the source they are citing in case you have not been able to look at it, you can see they are talking about Cauchy filters in the first sentence of the proof: