Let $X$ be a topological space, for $n \geq 1$ we define $F_n(X) = \lbrace{ (x_1,...,x_n) \in X^n | x_i \neq x_j for i \neq j \rbrace} $ the configuration space of n points of X.
My question is about the path-connected components of $F_n(\mathbb{R}^d)$.
I know that :
1- for $n=1$, $F_1(X) = X$ and hence $\pi_0(F_1(\mathbb{R}^d)) = \pi_0(\mathbb{R}^d) = \lbrace \mathbb{R}^d \rbrace \: \forall d \geq 1 $
2- $\forall n \geq 1$, $F_n(\mathbb{R}) = \bigcup \lbrace x_i < x_j \: or \: x_i > x_j \: | \: \forall \: 0 \leq i,j \geq n \rbrace $
Hence $\pi_0(F_n(\mathbb{R})) = \lbrace x_i < x_j \: or \: x_i > x_j \: | \: \forall \: 0 \leq i,j \geq n \rbrace $
I am unable to see how the path-connected componnents of $F_n(\mathbb{R}^d$) look like for all $n \geq 1$ and for all $d \geq 1$...
