Suppose that we have a first order language $\mathcal{L}$ that extends the language of rings. Let $T$ a be a topological $\mathcal{L}$-theory of fields in the sense of Pillay.. this means that not only is $T$ a theory fields but also there is an $\mathcal{L}$-formula $\phi(x,\bar y)$ such that for each model $\mathcal{M}\models T$ the set $\{\phi(\mathcal{M},\bar a)|\bar a\subseteq \mathcal{M}\}$ is a basis of neighborhoods of $0$ for a field topology.
Suppose I have models $\mathcal{M},\mathcal{N}\models T$ expanding fields $K,L$ respectively such that $\mathcal{M}\subseteq \mathcal{N}$. Denote the bases of neighborhoods of $0$ of $\mathcal{M}$ and $\mathcal{N}$ by $\mathcal{V}$ and $\mathcal{W}$ respectively. Suppose we have a subset $\widetilde{\mathcal{W}}\subseteq \mathcal{W}$ of neighborhoods that extend the sets in $\mathcal{V}$ in a way that is compatible with the field operations (to be more precise, see the definition 2.3 in this paper).
My question is, if I have a point in $t\in L$ such that $t\in W$ for all $W\in \widetilde{\mathcal{W}}$, then is it true that $t\in\phi(\mathcal{N},\bar c)$ for each $\bar c\subseteq \mathcal{M}$? In other words, if we have a point which is infinitesimally small with respect to $\widetilde{\mathcal{W}}$, is it necessarily contained in the canonical extensions of elements $\mathcal{V}$? (Note that above I have required $\widetilde{\mathcal{W}}$ is the set of the canonical extensions).
Thanks in advance for you time.
