In Segal's "Configuration Spaces and Iterated Loop Spaces", I'm not understanding the map $C_n(X) \rightarrow \Omega^n S^n X$ given in the first page, Section 1 (the picturesque electrostatic map) as well as the definition he gives on Page 218 in Section 3.
In particular, in Section 3, the definition of $D_n(X)$ isn't very clear to me, as well as $i_{\alpha}$ that he uses.Especially, this phrase "... disjoint open unit disks in $\mathbb{R}^n$..." is confusing me alot. Because disjoint open unit disks are a variable collection - there are so many of them. So we pick one such collection, and we're choosing a finite set (in the disjoint collection)? [But how are we choosing such a collection of disjoint open unit disks?]
But then, we can choose a finite set, and find a suitable collection of disjoint open unit disks such that the points in the finite set lies inside these open disks (since $\mathbb{R}^n$ is Hausdorff) - so isn't $C_n(X)$ the same as $D_n(X)$?
Maybe some good examples would clear things out.
Also, why is the map on Section 1 homotopic to the one given on Section 3?
Edit :As requested by Ryan Budney, I'm posting some relevant screenshots from the paper.
Definition of "Labelled by X"
Now, section 3...
