Uniform finiteness and viscerality

Let \(\mathcal {M}=(M,\cdots )\) $\mathcal{M}=(M,\cdots )$ be a first-order structure, \(\phi (x,y;\overline{z})\) $\varphi (x,y;\overline{z})$ and \(\psi (\overline{z},\overline{w})\) $\psi (\overline{z},\overline{w})$ be formulas. A visceral definable uniform structure on \(\mathcal {M}\) $\mathcal{M}$ is given by an uniformly definable family \({\mathcal {B}}_{\psi ,\overline{b}}=\{\phi (M^2;\overline{a}):\mathcal {M}\models \psi (\overline{a},\overline{b})\}\) ${\mathcal{B}}_{\psi ,\overline{b}}=\{\varphi (M^2;\overline{a}):\mathcal{M}\vDash \psi (\overline{a},\overline{b})\}$ such that \(\overline{b}\in M^{\mid \overline{w} \mid }\) $\overline{b}\in M^{\mid \overline{w}\mid }$ and it induces an uniform topology over M such that every infinite definable subset \(X\subseteq M\) $X\subseteq M$ has non-empty interior and each basic open set is infinite. In this note, we show that the structure \(\mathcal {M}\) $\mathcal{M}$ satisfies uniform finiteness and is the visceral definable uniform structure by \({\mathcal {B}}_{\psi ,\overline{b}}\) ${\mathcal{B}}_{\psi ,\overline{b}}$ if and only if every \(\omega \) $\omega$ -saturated elementary extension of \(\mathcal {M}\) $\mathcal{M}$ is the visceral definable uniform structure by \({\mathcal {B}}_{\psi ,\overline{b}}\) ${\mathcal{B}}_{\psi ,\overline{b}}$ .