Let $X$ be a compact metrizable contractible locally contractible topological space of finite covering dimension. Is $X$ homeomorphic to a CW complex?
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2$\begingroup$ As you probably know, it is an open problem whether a closed topological $4$-manifold is homeomorphic to a CW complex. I gather you are asking for a countexample in a larger class of spaces. Your assumptions imply that $X$ is an ANR.. $\endgroup$– Igor BelegradekCommented Aug 17, 2020 at 23:56
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2$\begingroup$ It seems the following is a counterexample: Let $X$ be the union of the straight line segments $I_n$ in $\mathbb R^2$ where $I_n$ joins the origin and the point $(1, 1/n)$, where $n$ is a nonnegative integer, and $I_0$ joins the origin and $(1,0)$. $\endgroup$– Igor BelegradekCommented Aug 18, 2020 at 0:09
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$\begingroup$ To show that this example has no CW structure note that the space is compact, so the complex is finite, which easily leads to a contradiction. $\endgroup$– Igor BelegradekCommented Aug 18, 2020 at 1:38
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$\begingroup$ also the compactification of the tree of which $F_2$ acts, is a counterexample. $\endgroup$– HenrikRüpingCommented Aug 18, 2020 at 3:58
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$\begingroup$ @IgorBelegradek I don't think it's an open problem whether a contractible manifold is homeomorphic to a CW complex. $\endgroup$– user145520Commented Aug 18, 2020 at 6:30
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