Abstract: Any free group can by given a non-discrete Hausdorff topology by a method exhibited by Marshall Hall, drawing on results by K. Iwasawa. Any group can be expressed as the quotient of a free group by a normal subgroup of that free group, so the topology which Hall imposes on the free group also provides a quotient topology on the selected group. We will first follow Hall's construction of the topology for the free group. Then we will relate group theoretical properties of that normal subgroup of the free group to properties of the quotient topology.