I am just learning what a topology is and from what I have understood up till now is that a topological space is nothing but a set with a notion of nearness that is given introducing open sets.
Ok, so, in the definition of manifold that I have seen (in Wald's general relativity book) a manifold is constructed mapping subsets of the manifold to be set with open subsets of $\mathbb{R}^n$.
This way we introduce a topology in our manifold to be, since we can use the notion of open balls in $\mathbb{R}^n$ to define open balls of the manifold and hence we get a topology.
Now, we can add more structure to the manifold endowing it with a metric and making it a metric space.
Now my question. I know that a metric induces naturally a topology. But we already had a topology before itroducing the metric structure, so, are this two topologies the same topology?
For the Riemannian case this is surely true, you may find it in "Foundations of Differential Geometry" by Kobayashi and Nomizu, volume 1, page 166, proposition 3.5.
In the pseudo-Riemannian case this is not true as indicated in a comment above by Zhen Lin (the topology induced by the pseudometric will no longer be Hausdorff, for instance).