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.



Properties of the class of topological spaces possessing a CW-structure
Ok, here is a simple example when this fails: let $X$ be $\mathbf{R}^2$ minus the origin and consider the automorphism of $X$ given by $(x,y)\mapsto (2x,y/2)$. This automorphism generates a group $G$. The quotient map $X\to X/G$ is a covering, but $X/G$ is not Hausdorff.


This answers the question as it is stated, but maybe you wanted $B$ to be Hausdorff (in which case I don't know what to do).


