i'm starting to study manifold in order to understand Einstein's relativity but there is something I don't understand. A manifold $M$ is defined as being locally homeomorphic to $\Bbb R^n$. 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. The question is: Are the topology of $M$ and $\Bbb R^n$ the same? Because then I put a distance in $\Bbb R^n$ and the distance will be locally the same in $M$. So since a distance induce a topology I think they must be the same.

Can I say that if $R^n$ is a metric space and $M$ is a manifold on this metric space then: The metric induce a topology on $R^n$, the metric of $R^n$ induce a metric on $M$ and so a topology?

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    $\begingroup$ A similar question was asked here. $\endgroup$ – Dietrich Burde Mar 12 '19 at 14:59
  • $\begingroup$ The metrics won't be the same globally: think of a solid donut sitting in 3-space. The metric inside the solid donut between points $P$ and $Q$ is given by measuring the length of the shortest curve joining $P$ and $Q$ (lying a bit here). Imagine that $P$ and $Q$ are on opposite sides of the donut hole. Their path won't be a straight line! $\endgroup$ – Prototank Mar 12 '19 at 19:30

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