If $U \subset Y \subset \mathbb{R}^n$, and $U$ is open relative to $Y$, then is $\Pi_k(U)$ open relative to $\Pi_k(Y)$? ($\Pi_k$ is a projection map)

Question

Here, $\Pi_k: \mathbb{R}^n \to \mathbb{R}^k$ denotes the projection map, mapping an element of $\mathbb{R}^n$ to its first $k$ coordinates. If $x = (x_1,\ldots,x_n)$, then $\Pi_k(x) = (x_1,\ldots,x_k)$.

To be precise, I am using the definition of "open relative" as defined in Rudin: Let $(X,d)$ be a metric space and suppose that $E \subset Y \subset X$. We say that $E$ is open relative to $Y$ if to each $p \in E$ there is associated an $r > 0$ such that $q \in E$ whenever $d(p,q) < r$ and $q \in Y$.