Let $Y$ be a closed subset of $\mathbb{R}^m$ (in fact $Y$ is convex and compact, but I think the extra assumptions are irrelevant). Let $A \in \mathbb{R}^{n \times n}$ be a non-singular matrix (so $A^{-1}$ exists). Let $C \in \mathbb{R}^{m \times n}$ be any matrix. Is the set $ Y' = \{ C A x \in \mathbb{R}^m \, : \, x \in \mathbb{R}^n, C x \in Y\} $ also closed?
Note: just to be clear, the definition of $Y'$ means $Y' = C A X = \{ C A x \in \mathbb{R}^m \, : \, x \in X \}$ where $X = \{ x \in \mathbb{R}^n \, : \, C x \in Y \}$.