Let $K$ be a global field and set $O := \prod\_{v\nmid \infty} O\_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}\_n(O) = \prod\_v \mathrm{GL}\_n(O\_v)$ with the product of the $v$-adic topologies and suppose that $H \le \mathrm{GL}\_n(O)$ is a subgroup of finite index. 


1. Is $H$ necessarily open?
2. Is $H$ necessarily a congruence subgroup? (I.e., does it contain $\mathrm{Ker}(\mathrm{GL}\_n(O) \rightarrow \mathrm{GL}\_n(O/NO))$ for some integer $N$?)


A positive answer to 2. would imply a positive answer to 1. 


I will accept a complete answer to either 1. or 2. as an answer. Also, I would be especially happy if you discussed a generalization of 1. and 2. for arbitrary reductive group schemes (or even a more general class of group schemes) over the ring of integers of $K$.



Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$
The answer to 1 is no. For example each $GL\_n(O\_v)$ has a continuous map to $\{ \pm 1\}$ so your product has a quotient isomorphic to $F\_2^{\mathbf{N}}$. Zorn's lemma will give you a non continuous linear form on that, and the pullback of the quotient is a non closed subgroup of index $2$. 


Edit: the pullback of the kernel, not of the "quotient"


