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at.algebraic topology – Covering picture of a linked CW-complex necessity not breathe a CW-complex

This query is already requested right here MSE, and there’s an respond primarily based on some surmise(in all probability quiet launch). I’m posting the identical query for a counterexample(if any, not primarily based on such surmise).

Problem:Let $X$ breathe a linked CW-complex, and $Y$ breathe a linked topological area. Suppose $p: Xto Y$ is a protecting map. Does there

live a CW-structure on $Y$? More typically, is $Y$ homotopically equal to a CW-complex?

Note that the opposite route is limpid: Any protecting of a linked CW-complex can at all times breathe given a CW-structure, lifting the traits map of cells of abject area such that the protecting map is a mobile map.

I consider that the respond to the above drawback is not any, however I’ve no counterexample.

$bullet$ Notice that $Y$ is regionally path-connected because the protecting map is an area homeomorphism, therefore $Y$ is path-connected too. So, we cannot deem areas ${0}cupleft{frac{1}{n}:ninBbb Nright}$ or Topologist Sine Curve as $Y$. Notice that each ${0}cupleft{frac{1}{n}:ninBbb Nright}$ or Topologist Sine round are usually not homotopically equal to a CW-complex.

$bullet$ Similarly, we cannot deem the Hawaiian Earring(this isn’t semi-locally merely linked) as $Y$: The linked CW-complex $X$ has the common mask in order that $X$ is semi-locally merely linked, however the property “semi-locally simply connected” is preserved below an area homeomorphism.

So, I’m speed out of examples. Any ameliorate will breathe appreciated. Thanks in forward.

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