Let $(G, \cdot)$ be a group for $a \in G$ let $U_{a}:=\left\{g a g^{-1} \mid g \in G\right\}$ ,
i) show that $\{ U_a \}_{a\in G}$ is a base for some topology on $G$.
ii) this topology is Hausdorff iff $G$ is an abelian group .
iii) when this topology is metrizable ?
For i The first property of base is essentially trivial. We need to show that any $a \in G$ belongs to at least one of those sets ( note that because $ a \in U_a $ ). Now for any $a \neq b$ we have $\mathrm{U}_{\text {a }} \cap \mathrm{U}_{\mathrm{b}}=\varnothing$ then this is a base for some topology on $G$ .for ii if $G$ be abelian group then for any $a \in G$ we have $U_a = \{a\}$ and any $U_a = \{a\}$ is open then this topology is hausdorff . For iii I think $G$ is metrizable iff $G$ be Abelian and meter is discrete metric because any single point set is open .