We show that, for a certain large class of power-bounded $o$-minimal $\mathcal{ L}_T$-theories $T$ whose field of exponents is infinite-dimensional as a vector space over the rationals, any definable set in a $T$-convex valued field $(\mathcal{ R}, \mathfrak{ O})$ is in a precise sense the limit of a family of $\mathcal{ L}_T$-definable sets indexed over the residue field. Alternatively, in the mainstream model-theoretic language, this says that if $(\mathcal{ R}', \mathfrak{ O}')$ is an elementary substructure of $(\mathcal{ R}, \mathfrak{ O})$ and if the residue field of $\mathfrak{ O}$ contains an element that is infinitesimal relative to the residue field of $\mathfrak{ O}'$ then any set $A \subseteq (\mathcal{ R}')^m$ definable in $(\mathcal{ R}', \mathfrak{ O}')$ is the trace of a set definable in $\mathcal{ R}$.
Categories: math.LO