This question is already asked here [MSE][3], and there is an answer based on some conjecture([probably still open][4]). I am posting the same question for a counterexample(if any, not based on such conjecture). 


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  > **Problem:** Let $X$ be a connected CW-complex, and $Y$ be a connected topological space. Suppose $p: X\to Y$ is a covering map. Does there
> exist a CW-structure on $Y$? Is $Y$ homotopically equivalent to a CW-complex?

Note that the other direction is clear: Any covering of a connected CW-complex can always be given a CW-structure, lifting the characteristics map of cells of base space such that the covering map is a cellular map. 

I believe that the answer to the above problem is no, but I have no counterexample.

$\bullet$ Notice that $Y$ is locally path-connected as the covering map is a local homeomorphism, hence $Y$ is path-connected also. So, we can not consider spaces $\{0\}\cup\left\{\frac{1}{n}:n\in\Bbb N\right\}$ or Topologist Sine Curve as $Y$. [Notice that both $\{0\}\cup\left\{\frac{1}{n}:n\in\Bbb N\right\}$ or Topologist Sine curve are not homotopically equivalent to a CW-complex.][1]

$\bullet$ Similarly, we can not consider the [Hawaiian Earring(this is not semi-locally simply connected)][2] as $Y$: The connected CW-complex $X$ has the universal cover so that $X$ is semi-locally simply connected, but the property "semi-locally simply connected" is preserved under a local homeomorphism.

So, I am run out of examples. Any help will be appreciated. Thanks in advance.


[1]:https://math.stackexchange.com/questions/523416/is-there-any-example-of-space-not-having-the-homotopy-type-of-a-cw-complex
 
[2]: https://math.stackexchange.com/questions/1104275/why-is-hawaiian-earring-not-semilocally-simply-connected





[3]: https://math.stackexchange.com/questions/3981109/covering-image-of-a-connected-cw-complex-need-not-be-a-cw-complex
[4]: https://mathoverflow.net/questions/73428/when-is-a-compact-topological-4-manifold-a-cw-complex