is a continuous function $h: S_{G}(M) \to C$ such that $f=h\circ r$, where $r\colon G^* \to S_{G}(M)$ is the obvious map.

We extend the notion of the definable flow in a natural way. Namely, a flow $(G,X)$ will be called [*$\Delta$-definable*]{} if for every $x \in X$ the map $f_x\colon G \to X$ given by $f_x(g)=gx$ is $\Delta$-definable.

The first part of the following observation was made in [@KrPi Remark 1.12], and the generalization to the second part can be obtained analogously.

\[definability of products\] i) A product of definable $G$-flows is a definable $G$-flow.\
ii) A product of $\Delta$-definable $G$-flows is a $\Delta$-definable $G$-flow.

Whenever we consider the quotient of $G^*$ by a bounded, $M$-invariant equivalence relation $E$, we can equip it with the [*logic topology*]{} which is defined by saying that a subset of the quotient is closed if its preimage in $G^*$ is type-definable (equivalently, type-definable over $M$). If $E$ is type-definable, then $G^*/E$ is a compact, Hausdorff space; if $E$ is only invariant, then $G^*/E$ is only quasi-compact. The definition of the logic topology applies in particular to the quotient $G^*/H$, where $H$ is an arbitrary bounded index, $M$-invariant subgroup of $G^*$. If $H$ is type-definable, 