then $G^*/H$ is a compact, Hausdorff group. For details on the logic topology see e.g. [@KrNe] and [@Pi1 Section 2]. The following basic remark will be useful later.

\[remark: denseness very basic\] Let $E$ be an bounded, $M$-invariant equivalence relation on $G^*$. Then $G/E$ is dense in $G^*/E$

Consider any non-empty, open subset $U$ of $G^*/E$, and let $\pi \colon G^* \to G^*/E$ be the quotient map. Then $\pi^{-1}[U]$ is a non-empty, $\bigvee$-definable over $M$ subset of $G^*$, and as such it has a point in $G$.

Universal $\Delta$-definable $G$-ambit {#section: definable ambit}
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From now on, we fix a set $\Delta$ of formulas about which we assume that:

1.  It consists of some formulas $\delta(x;y,z,\bar t)$ of the form $\psi(y\cdot x \cdot z; \bar t)$.

2.  The formula $G(y \cdot x\cdot z)$ is in $\Delta$, where $G(x)$ is a formula over $\emptyset$ which defines $G$.

This makes sure that any left or right translate of a $\Delta$-formula by an element of $G$ is still a $\Delta$-formula. Note that if all formulas as in (1) are included in $\Delta$, then being “a subset of $G$ which is $\Delta$-definable over $M$” is the same thing as being “a subset of $G$ 