Why the open-ball topology functor from $\boldsymbol{Met_c}$ to $\boldsymbol{Top_m}$ is not an isomorphism? Are they isomorphic at all?

Question

Let $\boldsymbol{Met_c}$ denote the category of metric spaces whose morphisms are all continous maps. And let $\boldsymbol{Top_m}$ denote the category of metrizable topological spaces whose morphisms are all continous maps. According to the book Abstract and Concrete Categories (The Joy of Cats), the functor from $\boldsymbol{Met_c}$ to $\boldsymbol{Top_m}$ that associate with each metric space its induced topological space is an equivalence but not an isomorphism. Why it is not an isomorphism? Are they isomorphic at all?

Answer 1

Different metrics on the same set can yield identical topologies (consider scaling a given metric). So the functor in question is not injective.