Small extensions of analytic fields

An extension $K/k$ of analytic (i.e. real valued complete) fields is called small if it is topologically-algebraically generated by finitely many elements. We prove that this property is inherited by subextensions and hence topological generating degree of such extensions is monotonic. Much more detailed results are obtained in the case of degree one. Let $k$ be an analytic algebraically closed field of positive residual characteristic $p$ and $K=\widehat{k(t)^a}$ with a non-trivial valuation. In a previous work it was shown that the set $I_{K/k}$ of intermediate complete algebraically closed subextensions $k\subseteq F\subseteq K$ is totally ordered by inclusion. In this paper we show that $I_{K/k}$ is an interval parameterized by the distance between $t$ and $F$. Moreover, logarithmic parameterizations induced by other generators differ by PL functions with slopes in $p^{\mathbb Z}$ and corners in $|K^\times|$, so $I_{K/k}$ acquires a natural PL structure.