# When we refer to topology on a metric space $S$ , do we mean the topology generated by open ball?

I've just learned a throrem which states that : A metric space has the structure of a topological space in which the open sets are unions of balls .

But the theorem only told me there "exist" one topology with respect to the metric.When we refer to topology on a metric space $$S$$ , do we mean the topology generated by open ball ?

Yes, a metric space $$(S, d)$$ induces a topology $$\mathcal{T}$$ which is generated by the basis $$\mathcal{B} = \{B(x, r) \ | \ x \in S \ \text{ and } \ r > 0\}$$
When authors refer to the topology on this metric space $$(S, d)$$, they usually mean the topology $$\mathcal{T}$$ above, which you can think of as the topology generated by open balls.
Yes. So you can say that a set $$U$$ is open iff for each $$x\in U$$, there exists an open ball $$B(x,r)$$ centered at $$x$$ with $$B(x,r)\subset U$$.