Question

Prove or disprove any metric defined on X(#0) induces a topology on X

Answer 1

Every metric space is a topological space in a natural manner.

The metric topology is the topology on $(X,d)$ generated by the base $B_r(x)=\{y\in X\colon d(x,y)<r\}$ (open balls). It is base because

1. Union of $B_r(x)$ cover $X$ (obviously)
2. $x\in B_{r_1}(x_1)\cap B_{r_2}(x_2)$ then $x\in B_r(x) \subset B_{r_1}(x_1)\cap B_{r_2}(x_2)$ with $r=\min\{r_1-d(x,x_1),r_2-d(x,x_2)\}$

A base generates a topology on $X$ that has, as open sets, all unions of elements of a base.