Assume $\mathcal{X},\mathcal{U}\subseteq \mathbb{R}^n$ are nonempty, closed convex, and let $$ \Omega=\{z:z\in \text{SOL}(\mathcal{X},Mx-u),u\in\mathcal{U}\}, $$ where SOL is the solution set of the variational inequality VI$(\mathcal{X},Mx-u)$ and $M$ is positive definite, i.e., $$ (y-z)^\top (Mz-u)\geq 0,\quad \forall y\in\mathcal{X}. $$ Is the set $\Omega$ convex and closed?
I know that for any given $u\in\mathcal{U}$, the set SOL$(\mathcal{X},Mx-u)$ is convex and closed since $\mathcal{X}$ is closed and $Mx-u$ is strongly monotone in $x$. But I am not sure whether $\Omega$ is convex closed since it is the union of infinitely many sets.
Another attempt could be using the fact that $u\mapsto z(u)$ is continuous. But again, I cannot guarantee that the set $z(\mathcal{U})$ is closed.
What is the best way to approach this problem?