f(x)) : x \in X \} 
\  \subseteq\  Y \times {{\mathbb C}}\ \ .$$

Each $\widehat f$ is compact.

We plan to apply the next definition and lemma to sets of the form $\widehat f$:

Fix $E \subseteq Y \times {{\mathbb C}}$ and $\Phi: {{\mathbb C}}^m \to {{\mathbb C}}$. Then $E_y = \{z : (y,z) \in E\}$ and $$\Phi * E \ =\  \bigcup_{y \in Y} \Phi( (E_y)^m ) \ \subseteq\  {{\mathbb C}}\ \ .$$

\[lemma-open-cover\] Suppose that $F \subseteq Y \times {{\mathbb C}}$ is compact and $\Phi: {{\mathbb C}}^m \to {{\mathbb C}}$ is continuous. Let ${{\mathcal B}}$ be an open base for $Y \times {{\mathbb C}}$ which is closed under finite unions. Then $\Phi * F$ is compact and $\Phi * F = \bigcap\{ \Phi * \overline U : U \in {{\mathcal B}}\ \&\ F \subseteq U \}$.

\[lemma-scat-phi\] Assume that $\pi: X \to Y$ is $n$–supertight and $f \in C(X)$. Fix a continuous $\Phi : {{\mathbb C}}^{n} \to {{\mathbb C}}$ such that $\Phi(z_1, \ldots, z_n) = 0$ unless all $n$ of the $z_1, \ldots, z_n$ are different. Then $\Phi * \widehat f$ is compact and countable, and hence scattered.

Compactness follows from the compactness of $X,Y$. By $n$–tightness, 