Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).

Is it true that for a covering map $E\stackrel{f}{\to} B$ with $E\in{\mathcal C}$ we have $B\in{\mathcal C}$, too?

It *is* true that the total space of a covering lies in ${\mathcal C}$ if the base space does, but the reverse implication is not clear to me.

**Edit**

As Algori pointed out, the quotient space is not even Hausdorff in general. What about finite regular coverings, i.e. those which come from a free action of a finite group on the total space? Is it true then that the quotient space carries a CW-structure, too?

I'm interested in that because this would imply that given a free group action of a finite group on a "nice" space like a CW-complex, one can always choose a CW-structure with respect to which $G$ just permutes cells. Then the corresponding cellular complex would be a (possibly nice) complex of ${\mathbb Z}G$-modules (for example, if the space was a sphere, then this procedures can be used to construct a periodic ${\mathbb Z}G$-resolution of the trivial module ${\mathbb Z}$, showing that the group has to have periodic invariants like homology and cohomology; in this particular case, however, things behave well as the quotient space ${\mathbb S}^n/G$ is still a compact manifold).

Thank you.