Pushfoward of a CW complex structure by a covering map

Question

Let $p:Y\to X$ be a covering map. If $X$ has a CW complex structure, then we can give a CW complex structure on $Y$ so that $p$ becomes a cellular map, by lifting the characteristic maps (cf. Euler characteristic of covering space of CW complex). I am curious about the converse. Suppose $Y$ is a CW complex. Then can we give $X$ a CW complex structure?

Answer 1

The action of $\mathbb{Z}$ on $\mathbb{R}^2\setminus\{0\}$ generated by $(x,y)\mapsto(2x,y/2)$ induces a covering map $\mathbb{R}^2\setminus\{0\}\rightarrow(\mathbb{R}^2\setminus\{0\})/\mathbb{Z}$ whose base is not Hausdorff, hence cannot be a CW-complex. (This is Exercise 1.3.25 from Hatcher's Algebraic Topology).