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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.

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  • $\begingroup$ Your reference to Definition 2.3 of the Guzy-Point paper isn't clear to me as the definition isn't about compatibility with the field operations but about elements of the bigger model being infinitely close from the point of view of the smaller one. Do you mean Definition 2.2 (2)? $\endgroup$
    – user12283
    Nov 29, 2015 at 8:48

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