Let $C_n$ be the set of unordered distinct $n$-tuples in $\mathbb{R}^2$: $$C_n= \left\{ (x_1,\ldots,x_n)\in (\mathbb{R}^2)^n \mid x_i\neq x_j \text{ for }i\neq j \right\}/S_n .$$

I wish to prove that its fundamental group $\pi_1(C_n)=B_n$, where $B_n$ is the Artin braid group on $n$ strands.

While there are probably various standard techniques for computing fundamental group, I do not know which I should look out for. Therefore, I am looking for a sketch of the proof, or a list of key concepts/theorems I should read up on.