I am doing work with model theory and came across the following exercise:
"Consider a set $Y$ in some structure $\mathcal{M}$. Let $f:Y\to C$ be a definable map to a compact space $C$, with dense image. Suppose $Y\subseteq Y^{*}$. Then there is a unique extension of $f$ to $Y^{*}$, called $f^{*}$, that is M-definable. $f^{*}(a) = \cap_{\alpha \in p} \overline{f(\alpha(M))}$ where $p = tp(a/\mathcal{M})$.
Prove that this extension is surjective. "
So my thought to use the dense image of $f$ since we know that the preimage of dense set under an onto map will be dense also. But I was unable to show that this would be the case.
Outline of my unsuccessful attempt. I don't know if this can work, or whether need to start over:
I try to use the definable map and $\mathcal{M}$-definable map properties. So if $f$ is definable, we can find a definable set $Y_{0}\subseteq Y$ such that for disjoint closed sets $C_{1},C_{2}\subseteq C$ (which will also be compact) $C_{1}\subseteq Y_{0}$ and $C_{2}\cap Y_{0} = \emptyset.$ For an $\mathcal{M}$ definable map, like our $f^{*}$, preimage of closed set is type definable. So maybe I show that the whole of $Y^{*}$ is dense?
Thank you for help.