Suppose $X$ and $Y$ are compact Hausdorff spaces. Let $\varphi\colon C(X)\to M_{n}(C(Y))$ be any $*$-homomorphism. If $\pi$ is an irreducible representation of $M_{n}(C(Y))$, then $\pi$ is unitarily equivalent to a point evaluation $\textrm{ev}_{y}$. The $*$-homomorphism $\textrm{ev}_{y}\circ\varphi\colon C(X)\to M_{n}(\mathbb{C})$ is a representation of $C(X)$. Since it's a finite-dimensional representation, we can find a unitary $u_{y}\in M_{n}(\mathbb{C})$ and a set of points $X_{y}=\{x^{y}_{1},\ldots,x^{y}_{n}\}\subset X$ such that for all $f\in C(X)$,
$$
(\varphi\circ f)(y)=(\textrm{ev}_{y}\circ\varphi)(f)=u_{y}
\begin{pmatrix}
f(x^{y}_{1}) & 0 & \cdots & 0\\
0 & f(x^{y}_{2}) & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & f(x^{y}_{n})
\end{pmatrix}
u_{y}^{*}.
$$

**My question is:**

> Is the set $\widetilde{X}:=\bigcup_{y\in Y}X_{y}$ closed in $X$?

This question is in a similar vein to one of my earlier questions: https://mathoverflow.net/questions/306383/closeness-of-points-in-the-irreducible-decomposition-of-a-c-algebra-repres