It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff it contains an open subgroup. This suggests making a topology on the modular group $\Gamma$ with the subgroups $\Gamma(N)$ as a basis of open neighborhoods of the origin so that $\Gamma$ becomes a topological group. It would then follow that a subgroup of $\Gamma$ is a congruence subgroup iff it is open.


Furthermore, for any $\gamma \in \Gamma$ not equal to the identity, there exists $N$ such that $\gamma \notin \Gamma(N)$, so this topology is Hausdorff, even totally disconnected.


I was inspired in part by [this thread](https://mathoverflow.net/questions/20929/distinguishing-congruence-subgroups-of-the-modular-group) and looked at [this paper](http://www.ams.org/journals/proc/1996-124-05/S0002-9939-96-03496-X/S0002-9939-96-03496-X.pdf) but could not find anything about this idea.


Has anyone considered this topology? Does it provide insight into the problem of determining whether a group is a congruence subgroup?



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](https://mathscinet.ams.org/mathscinet-getitem?mr=0272790) (42 #7671) 14.50,
Serre, Jean-Pierre,
Le problème 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. 


