Equivalent definition of residually finite group

Question

I'm reading this paper that deals with group $C^*$-algebras and the definition of a residually finite group is if it has a separating family of finite index normal subgroups. I've seen many of the equivalent definitions online but none of them to be phrased in terms of a separating family, and I cannot seem to find a definition of that either.

Answer 1

Every group $G$ may be made into a topological group by taking as a basis of open neighborhoods of the identity, the collection of all normal subgroups of finite index in $G$. The resulting topology is called the profinite topology on $G$. A group is residually finite if, and only if, its profinite topology is Hausdorff. So any two points can be "separated" by disjoint open neighborhoods.