The fundamental group of a wedge sum of countably many circles is countably generated, hence countable.

Question

The fundamental group of a wedge sum of countably many circles is countably generated, hence countable.

I don't really understand this statement. How can I see that the free product of countable $\mathbb{Z}$'s is countable?

Answer 1

Let $G$ be a group generated by a countable set $S$ (and assume for simplicity that $S$ is symmetric). Then every element of $G$ is a finite product of elements of $S$. So there is a surjective function $\bigcup_{n\in \mathbb{N}} S^n \to G$ given by multiplication.

Since each $S^n$ is countable, their countable union is also countable, and so is $G$.