How metric on tangent space affects topology

Question

Hallo I'm starting to study manifold in order to understand Einstein's relativity but there is something I don't understand. I already asked something very similar today but now I have another question. A manifold $M$ is defined as being locally homeomorphic to $\Bbb R^n$ (is this just $\Bbb R^n$ with a topology or has it to be the Euclidean space?). 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. In the case I'm studying they use the tangent vectors in a point as basis of the tangent space and then they endow the tangent space with a norm so now the measure of $dx$ is defined. Integrating I have the distance between any two point of the manifold. What I don't understand is: if the metric induce a topology on the manifold is this topology the same as before? is the space still in homeomorphic to $\Bbb R^n$?

Answer 1

Yes, the metric in the tangent space induces the same topology. Note that this is not a complete coincidence, because it is a requirement that the metric in the tangent space varies at least continuously with the point of the manifold.