Exercise 11, § 8

Let $X$ be a topological space. Let $Z$ denote the product space $\mathbf{R}^X$, and for each $x \in X$ let $Z_x$ be the subset of $Z$ consisting of all $u \in \mathbf{R}^X$ such that $u(x) \in \mathbf{Q}$ and $u(y) \notin \mathbf{Q}$ for all $y \in X$ other than $x$. Let $Y$ be the subspace of the product $X \times Z$ which is the union of the sets $\{x\} \times Z_x$ as $x$ runs through $X$. Show that $Y$ is Hausdorff and that $X$ is homeomorphic to a quotient space of $Y$.