# C-minimality and ultrametrics

Consider $\DeclareMathOperator{\bfT}{\mathbf{T}}$ the theory $\bfT_{\operatorname{um}}$ of ultrametric spaces, where the language $L$ has two-sorts $VF$ and $\Gamma_\infty=\Gamma \cup \lbrace\infty\rbrace$ and a binary relation $\lt$ and the function symbol $\DeclareMathOperator{\val}{val}$ $\val: VF \rightarrow \Gamma_\infty$. $\Gamma$ is an ordered divisible abelian group and $VF$ is a valued field. Let $T$, $\bfT_{\operatorname{um}}\subset T$ be a C-minimal theory and let $(VF,\Gamma)$ also denote a model of $T$. Then,

1. $\Gamma$ is o-minimal, and
2. for any closed $\alpha$-ball $C$ ($= \lbrace x \in VF \mid \val(x-y) \geq \alpha\rbrace$ for some $y$) the

set of open $\alpha$-subballs ($=\lbrace x \in VF \mid \val(x-y) > \alpha\rbrace$ for some $y$) of $C$ is strongly minimal and it acquires the structure of an affine space over the residue field $\mathbf{k}$. Please help me understand and prove (1) and (2). Thank you

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Seems that you are a few $short. Perhaps selling some kidneys on the black market can help... – Asaf Karagila Sep 13 '11 at 22:26 I'm not sure this question makes sense as written. Where does$T$(the C-minimal theory) enter into anything? – Noah S Sep 14 '11 at 3:31$\Gamma$and$VF$means also a model of$T$in the language$L$. – user16974 Sep 14 '11 at 8:34 You need to work on the statement of the question. You have a theory$T$and a theory$T_{um}$. You mention$T_{um}\$ in the first question, but then never again. Some people familiar with valued fields certainly know what you are asking, but the question is currently poorly stated. Also, aren't questions like this likely to be answered readily by looking in the Haskell, Hrushovski, McPhereson book? Or basic course notes on model theory of valued fields? –  James Freitag Sep 14 '11 at 17:21
Sorry, when I said first "question" I meant first "sentence" above. –  James Freitag Sep 14 '11 at 17:22
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