I was reading Is this proof that all metric spaces are Hausdorff spaces correct? to see how the statement is proved, but I have a question. Is it possible to generalize that any ball in metric space is open? How do we know the "open set" in metric space? Is it a convention to say that topology of metric space is topology generated by every ball around every point? I can’t think it right now but would there be any topology in metric space that some ball around some point is not open?