Extension of Galois Correspondence to Solenoids It is well-known that covering spaces of a sufficiently nice space $X$ are in bijective correspondence with subgroups of $\pi_1(X)$. However, "limiting" phenomena do not play nice with this correspondence. For example, if I take the double cover of the circle, then, topologically, this is just another circle. But it is more instructive to think of the two sheeted cover as an actual cover, with monodromy data telling me that when I pass around a loop, I flip to the other sheet.
One can iterate this construction on the resulting circle, and in fact can do this infinitely. You can ask for the projective limit of the system of covering spaces, each of degree $2$. This is not a covering space anymore, as the fiber is not discrete anymore, but is in fact homeomorphic to a Cantor set(!), which can be seen by showing it is homeomorphic to the ends of the infinite binary tree.
This viewpoint can give us a heuristic for why the space should not be path-connected, which you could probably turn into a proof if you try. The idea would be that two generic points of the solenoid live on sheets indexed by points in the Cantor set, which in general, involve traversing something like $2^k$ loops in order to reach the correct sheet. On the other hand, the solenoid is compact, so if I take any sequence of points $x_k$ differing by $2^k$ sheets, there is a limit point, but there shouldn't be a path between them, because it is "too long" to reach. It's rather like running from one end of the infinite binary tree back to finite space and then running to a different end. You just can't do this with a closed interval.
Is there some notion of "completion" of the fundamental group, in this case just $\mathbb{Z}$, where subgroups of this new object correspond to such whacked out "coverings?" It would be most interesting if a solenoid where you always take the $p$-fold cover "corresponded" to the $p$-adic integers, but let's not yet dare to dream too much.
 A: Formally what should happen if everything is sufficiently nice, and I don't know what is required to make everything sufficiently nice, is that the category of such pro-(finite coverings) should be equivalent to the pro-category of the category of finite coverings in the usual sense. If that's true, then by abstract nonsense it's equivalent to the category of continuous actions of the profinite completion of the fundamental group $\widehat{\pi_1(X)}$ (assuming that $X$ is path-connected and so forth as usual). This object famously shows up in the theory of the etale fundamental group if $X$ happens to be homotopy equivalent to the space of complex points of a smooth projective variety (edit: actually apparently the result holds for $X$ a connected scheme locally of finite type over $\mathbb{C}$).
The profinite completion of the fundamental group $\mathbb{Z}$ of the circle is the profinite integers $\widehat{\mathbb{Z}}$, which are isomorphic to a product $\prod_p \mathbb{Z}_p$ over all the $p$-adic integers (by the Chinese remainder theorem), and I guess that the $p$-adic solenoid should correspond exactly to the quotient map $\widehat{\mathbb{Z}} \to \mathbb{Z}_p$ although I haven't thought it through carefully. I'm more used to thinking of the $p$-adic solenoid in terms of Pontryagin duality.
