\[definable functions\] Let $C$ be a compact space. A map $f\colon G \to C$ is [*$\Delta$-definable*]{} if for all disjoint, closed subsets $C_1$ and $C_2$ of $C$ the preimages $f^{-1}[C_1]$ and $f^{-1}[C_2]$ can be separated by a $\Delta$-definable subset of $G$.

Now, we recall the definition of a definable map defined on the monster model and we extend it to local versions.

Let $C$ be a compact space.\
i) A function $f\colon G^* \to C$ is [*$M$-definable*]{} if for every closed $C_1 \subseteq C$ the preimage $f^{-1}[C_1]$ is type-definable over $M$.\
ii) A function $f\colon G^* \to C$ is [*$\Delta$-definable over $M$*]{} if for every closed $C_1 \subseteq C$ the preimage $f^{-1}[C_1]$ is $\Delta$-type-definable over $M$ (i.e. the intersection of sets defined by $\Delta$-formulas over $M$).

The first item of the next fact is [@GiPePi Lemma 3.2], and the second one (generalizing the first one) can be proved analogously. As usual, $C$ denotes a compact, Hausdorff space.

\[prolongation\] 1) Definable context:\
i) If $f\colon G \to C$ is definable, then it extends uniquely to an $M$-definable function $f^*\colon G^* \to C$. Moreover, $f^*$ is given by the formula $\{f^*(a)\}=\bigcap_{\varphi \in \operatorname{{tp}}(a/M)} \operatorname{{cl}}(f[\varphi(M)])$.\
ii) Conversely, if $f^*\colon G^* \to C$ is 