Question

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

Answer 1

disprove:

for a set A such that |A| >1, for a,b$\in$ A

let a is not equal to b and d(a,b)=m , then,

for the ball B(m/2,a) which has radius m/2 does not contain 'b', that is

b $\notin$ B(m/2,a)

this implies B(m/2,a) is neither equal to $\varnothing$ nor equal to set A

that means it does not induce a trivial topology.