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A manifold $M$ is defined in particular as being locally homeomorphic to $\mathbb{R}^{n}$. Homeomorphisms can be defined in terms of how they map open sets, namely an homemorphism $f$ and its inverse $f^{-1}$ have to map open sets to open sets. Then, it looks like the definition of manifold depends on which topology you use in $\mathbb{R}^{n}$. What I mean is that a local map from $M$ to $\mathbb{R}^{n}$ may or may not be an homeomorphism depending on which topology one uses in $\mathbb{R}^{n}$. This indeed looks very weird. What am I missing?

Thanks.

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    $\begingroup$ The point is that when you define manifolds you use $\mathbb{R}^{n}$ endowed with the standard topology. $\endgroup$
    – Dario
    Feb 23, 2015 at 9:12
  • $\begingroup$ But in principle one could use a different topology to define a manifold, is that right? Is there an example of such construction? $\endgroup$
    – Bilateral
    Feb 23, 2015 at 9:53
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    $\begingroup$ In principle one can use different topologies, but the point of defining manifolds is to have spaces that locally resembles a space with whom you are familiar, and topologically you are familiar with $\mathbb{R}^n$ with the standard topology. The fact is that euclidean $\mathbb{R}^n$ has a lot of properties that are well known and understood, while this is in general not true for other topologies. Moreover one would like to do calculus on manifolds and you basically know how to do calculus in the standard $\mathbb{R}^{n}$. $\endgroup$
    – Dario
    Feb 23, 2015 at 10:02

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