Fundamental group of countably many holes and one limit point

Question

So I proposed this Fundamental group of a space of infinite genus and an accumulation point same question to my professor in Complex Analysis when we were going over the fundamental group and homotopies. After class I asked him what he thought

$$\pi_1(\Bbb C \setminus (\{0\}\cup_{n\in\Bbb N} \{ \frac 1 n \}))$$

and he said he thought it would be

$$F_\omega = F_{\Bbb N}$$

the free group on countably many generators. It seems intuitive to me that this group should be different than

$$\pi_1(\Bbb C \setminus \Bbb N)$$ because you cannot "wrap" around the point at infinity in this set. So can anybody settle this and tell me which more commonly dealt with groups these fundamental groups are isomorphic to? Also, what would

$$\pi_1(\Bbb C \setminus \Bbb Q)$$ be isomorphic to, or

$$\pi_1(\Bbb C \setminus \left( \Bbb Q\times \Bbb Q \right) ) $$

(where I'm sure you guys can decipher my slight use of shorthand...)

Answer 1

A basic application of the classification theory for surfaces is that every connected, noncompact surface is homotopy equivalent to a graph, and so its fundamental group is free. The rank of that free group is finite if and only if the surface is homeomorphic to a compact surface minus finitely many points, and otherwise it is countably infinite. So for your example, the rank is indeed countably infinite.

Answer 2

I gave hints for a down-to-earth answer in the comments.

Here is the general picture: by a surface I will mean a connected, second-countable Hausdorff space which is locally homeomorphic to $\mathbb{R}^2$. It is a known classical fact that the fundamental group of any non-compact surface is a free group on at most countably many generators. (The countability follows for instance from building your surface as a cell complex with only countably many cells.) Further, if you take an already noncompact surface and remove a point from it, this has the effect of adding one more free generator to the fundamental group.

It follows that any surface which can be obtained from another surface by removing any closed (not necessarily discrete) countably infinite set of points has fundamental group a free group on a countably infinite set of generators. The OP is asking about two surfaces with this property.

With respect to the other question I will remark only that $\mathbb{C} \setminus \mathbb{Q}$ and $\mathbb{C} \setminus (\mathbb{Q} \times \mathbb{Q})$ are not surfaces, so the above considerations do not apply.