Uniqueness of Polish group topology

We show the limits of Mackeyʹs theorem applied to identity sets to prove that a given group has a unique Polish group topology. sets in Abelian Polish groups and full verbal sets in the infinite symmetric group are Borel. However this is not true in general. sh group with a neighborhood π-base at 1 of sets from the σ-algebra of identity and verbal sets has a unique Polish group topology. It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.