Could you explain how to show 9.6 in detail?

Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the same topology, yet for each x, y E X, d(x, y) 0} forms a basis for a topology on X. The topology generated by a metric d on X is called the d-metric topology for X. Definition. A topological space (X, J) is a metric space or is metrizable if and only if there is a metric d on X such that J is the d-metric topology. We sometimes write a metric space as (X, d) to denote X with the d-metric topology