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I've seen a definition of a manifold as a set $M$ with an atlas, that is, a collection of pairs $(U_\alpha,\phi_\alpha)$ where all $U_\alpha$ cover $M$ and the coordinate maps $\phi_\alpha$ are compatible. The only notion of topology in this definition is that $\phi_\alpha(U_\alpha)$ is an open set in $\mathbb R^n$ (I guess using the standard topology). In this definition we could identify $U_\alpha$ as an open set, so this defines a topology in $M$.

But most definitions require $M$ to have a topology previously defined so $U_\alpha$ are open sets from the beggining.

How are these two definitions related? It seems that the second definition allows many possible manifolds if we change the topology. Maybe the former is a particular case of the latter?

jinawee
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  • If you do that, you need to put some other hypothesis to ensure that the resulting space is locally euclidean, Hausdorff and has a countable basis. Given that, everything is ok. – Mariano Suárez-Álvarez Oct 02 '16 at 18:58
  • We need, for the atlas to produce a manifold, that the maps $tphi_\alpha $ are homeomorphisms, so I'm not on board with the claim that the "only notion of topology" here involves $n$-space. – hardmath Oct 02 '16 at 19:06

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