Definition (Metric Topology)

Let $(X,d)$ be a Metric Space. Let $$\mathscr{T}_{d}=\{ A\subseteq X:\forall a\in A,\,\exists r>0\ s.t. \ B(a,r)\subseteq A \}$$where $B(a,r)=\{ b\in X:d(a,b)<r \}$. We refer to $\mathscr{T}_{d}$ as the metric Topology on $X$ generated by $d$.