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

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.

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    $\begingroup$ If $Y$ is locally-compact then it will be a metrizable ANR, hence, homotopy-equivalent to a CW complex. $\endgroup$ Jan 13, 2021 at 16:24
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    $\begingroup$ I think an example is given in Exercise 1.3.25 of Hatcher. There is a covering space action of Z on the punctured plane with non-Hausdorff orbit spaces (call it Y) with fundamental group Z^2. Any map from Y to a Hausdorff space factors through a Hausdorffizafion Haus(Y). This space Haus(Y) is homeomorphic to S^1 x X, where X = R sqcup_0 R (the literal letter X without its endpoints). This has fundamental group Z. Hence any map from Y to a CW complex induces a map on pi_1 with cyclic image; there is no CW complex Y' with f: Y -> Y' which is an isomorphism on pi_1, as Z^2 is not cyclic. $\endgroup$
    – mme
    Jan 14, 2021 at 6:13


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