means that $gx_0$ tends to $p$).

Finally, we leave as an exercise (using nets and limits) to check that $*$ is associative.

Next, we give an explicit formula for $*$, which is similar to the one in the externally definable case.

\[proposition: formula for \*\] For any $p,q \in S_{G,\Delta}(M)$, $p/E_\Delta * q/E_\Delta=\operatorname{{tp}}_\Delta(a\cdot b/M)/E_\Delta$, where $b \models q$ and $a$ realizes a $\Delta$-coheir extension of $p$ over $M,b$ (i.e. $\operatorname{{tp}}_\Delta(a/M,b)$ is finitely satisfiable in $M$).

Note that a basis of open neighborhoods of $\operatorname{{tp}}_{\Delta}(ab/M)/E_\Delta \in S_{G,\Delta}(M)/E_\Delta$ consists of the sets

$$U_\varphi:=\{ r/E_\Delta : [r]_{E_\Delta} \subseteq [\varphi]\}$$ with $\varphi$ ranging over all $\Delta$-formulas over $M$ such that $\operatorname{{tp}}_{\Delta}(ab/M)/E_\Delta \in U_\varphi$, where $[r]_{E_\Delta}=\{ q \in S_{G,\Delta}(M): E_\Delta(r,q)\}$ and $[\varphi]=\{ q \in S_{G,\Delta}(M) : \varphi \in q\}$. Indeed, first of all, each set $U_{\varphi}$ is open in the quotient topology, because the preimage of the complement of $U_{\varphi}$ under the quotient map consists of all the types $r \in S_{G,\Delta}(M)$ for which there exists a type $r' \in S_{G,\Delta}(M)$ such that $E_\Delta(r,r')$ and $\neg \varphi(x) \in r'$ (and so we see that this preimage is closed as a projection of a closed set in a product of compact, Hausdorff spaces). Secondly, take any 