TITLE: Are the topology of a manifold and the topology induced by the metric of a manifold the same?
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: 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).