This is a question about the paper Omitting types and the Baire category theorem (Christopher J. Eagle, Franklin D. Tall).
We are given a contravariant functor $S$ from the category of finite sets to the category of topological spaces and open maps, satisfying some amalgamation property (dfn 1.1 page 2). The space $S(n)$ is to be thought of as the space of $n$-pointed models modulo elementary equivalence for some logical theory (aka $n$-types).
The space $S_{\omega}$ is the space of $\omega$-types (dfn 1.2 page 3, constructed as a filtered colimit) and they identify the space $S_{\mathcal{W}}$ of countable models with a subspace of $S_{\omega}$ (dfn 2.3 page 7). In lemma 2.3 page 11, it is said (or implied) that this definition gives $S_{\mathcal{W}}$ as an intersection of open sets. This is what I don't understand.
If $S$ takes values in Stone spaces, I can fill the gaps, but otherwise there seems to be some problems. In definition 2.3 page 7, I think they forgot to ask about $g$ that $a_i = b_{g(i)}$ as in definition 1.3 page 3, but this is not the problem. They also forgot to say that $U$ must be non empty, and I think this is what generates the problem.
Here is a reformulation of the definitions. A model of $S$ with underlying set $X$ is to be thought of as a natural transformation $X^n → S_n$ verifying some amalgamation property. This amalgamation property translates on the elements $\sigma∈S_{\omega}$ as follows (*). Let $f:n→m$ be a function.
$$\require{AMScd} \begin{CD} \omega^m @>{u}>> S_m\\ @VV{\omega^f}V @VV{Sf}V\\ \omega^n @>{v}>> S_n \end{CD}$$
The function $v$ sends $\alpha:n→\omega$ to $(S\alpha)(\sigma)$ and similarly for $u$. We say that $\sigma∈S_{\mathcal{W}}$ if for each function $f:n→m$, each open set $U⊆S(m)$ and each $\alpha∈\omega^n$, whenever $v(\alpha) ∈ (Sf)(U)$, there exists $\beta∈\omega^m$ with $\omega^f(\beta) = \alpha$ and $u(\beta) ∈ U$. We can restrict the quantification over $U$ by looking at only some basis of open sets.
The condition that $U$ must be nonempty in definition 2.3 corresponds to the condition "whenever $v(\alpha) ∈ (Sf)(U)$" above. We can rewrite the condition for $\sigma∈S_{\mathcal{W}}$ as a conjunction of open conditions on $\sigma$ if we suppose that $(Sf)(U)$ is closed for all $U$ in the basis of open sets, by writing the implication "whenever $v(\alpha) ∈ (Sf)(U)$, we have $B$" as $\lnot [(S\alpha)(\sigma) \in (Sf)(U)] \lor B$.
Did I miss something? How to obtain this lemma 2.13 page 11 (which is said to be immediate)?
(*) Elements of $S_{\omega}$ correspond to natural transformations $\omega^n → S_n$ by an application of the Yoneda lemma (and a bit more).