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For any locally Euclidean Hausdorff 2nd countable topological space $M$, can $M$ always be made into a topological manifold (not necessarily of some uniform dimension $n$ ... but if ever I assume various $n$'s depend on the connected component not the atlas) i.e. there always exists a continuous $C^0$ atlas for $M$? (I don't define topological manifold as merely a locally Euclidean Hausdorff 2nd countable topological space.)

I think this is an assumption sorta made in the ff question: Can manifold subsets always be made into submanifolds?

BCLC
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  • Please write down the definition of local euclidean and the existence of $C^0$ atlas. – Arctic Char Jun 13 '24 at 17:07
  • @ArcticChar 1- for every $p$ there exists a chart. (oh for the dimension thingy. i edited question.) 2- same as smooth atlas but change smooth to continuous ? – BCLC Jun 13 '24 at 17:27
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    For each $p \in M$ pick a neighborhood $p \in U_p \subseteq M$ and a homeomorphism $\psi_{U_p}\colon U_p \to \mathbb{R}^n$. Convince yourself that the collection of all $(U_p, \psi_{U_p})$ is a $C^0$-atlas for $M$. Conclude that $M$ has a unique maximal $C^0$-atlas and that there is little point in talking about $C^0$-structures in the first place. – Ben Steffan Jun 13 '24 at 17:44

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