How to Prove that metric space is topological space?

To prove that a metric space ((X, d)) is a topological space, you need to show that the open sets defined by the metric (d) satisfy the axioms of a topology. Specifically, you can define the open sets as the collection of all unions of open balls (B(x, r) = {y \in X \mid d(x, y) < r}) for all (x \in X) and (r > 0). Then, verify that this collection includes the empty set and the whole space (X), is closed under arbitrary unions, and is closed under finite intersections. If these conditions hold, then the metric space indeed induces a topology.