Basically the question above. Does any topological manifold admit a CW -structure?
If not, what assumptions make this a true statement?
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3Does this answer your question? Is every open topological $n$-manifold homeomorphic to a CW complex? $(n \geq 5)$ Generally yes for dim $\neq 4$, while for dim $4$ it is an open problem AFAIK. – freakish May 26 '23 at 13:49