Topology induced by a metric?
When we say that a metric space (X,d) induces a topology or "every metric space is a topological space in a natural manner" we mean that:
A metric space (X,d) can be seen as a topological space (X,τ) where the topology τ consists of all the open sets in the metric space? Which means that all possible open sets (or open balls) in a metric space (X,d) will form the topology τ of the induced topological space? Is that correct? 
Re: Topology induced by a metric?
Well, yes, but what are the open sets in (X,d)?
They are those sets A such that for every a in A, there is a r>0 such that B(a;r)={x in X : d(x,a)<r} is entirely contained in A. You can verify that these sets form a topology on X. 
Re: Topology induced by a metric?
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Re: Topology induced by a metric?
Oh I see, that's because the collection of open balls is a subset of the collection of all open sets.
It makes sense, thank you both for your time! 
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