Assume $A, B, C$ are profinite groups and $0\to A\to B\to C\to 0$ is an exact sequence of continuous maps. Which of the following assertions follows?:
(i) the subspace-topology induced on $A$ via $A\hookrightarrow B$ agrees with the given one.
(ii) the quotient-topology induced on $C$ via $B\twoheadrightarrow C$ agrees with the given one.