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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 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.

• 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.

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

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.

• 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.

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

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 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.

$\bullet$ 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.

$\bullet$ Similarly, we can not consider the Hawaiian Earring(this is not semi-locally simply connected) 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.

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 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.

• 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.

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

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 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 $Y$? More generally, 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.

$\bullet$ Similarly, we can not consider the Hawaiian Earring(this is not semi-locally simply connected) 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.

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 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.

$\bullet$ 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.

$\bullet$ Similarly, we can not consider the Hawaiian Earring(this is not semi-locally simply connected) 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.