Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology

To expand Henry Wilton's concise answer, the Congruence Subgroup Problem has a distinguished history including important work by Serre and a number of others (exploiting effectively the congruence topology). See for example:
MR0272790 (42 #7671) 14.50,
Serre, Jean-Pierre,
Le probl`eme des groupes de congruence pour SL2. (French)
Ann. of Math. (2) 92 1970 489–527.

This sort of topology on a group originates earlier, but the application here is highly original.

ADDED: Like many other journal articles, the one mentioned here by Serre is available in PDF format but only through JSTOR (or other library resource). There is a lot of literature, including my 1980 Springer Lecture Notes 789 Arithmetic Groups which cover some of the background as well as an expository account of Matsumoto's thesis.