A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ covers $X$.
Given any space $X$, is there a $D$-space $X_D$ and a continuous map $\varepsilon_D: X\to X_D$ such that for every continous map $f: X\to Z$ from $X$ to a $D$-space $Z$ there is a unique continous map $f_D: X_D \to Z$ such that $f =f_D \circ \varepsilon_D$?
