Given a manifold $M$, the Fadell’s configuration space (in topology called simply configuration space) is the manifold of $N$-tuples of pairwise distinct points in $M$.
It is important in the study of topological fibrations, in the study of arrangements of hyperplanes, of Knizhnik-Zamolodchikov connections and in study of geometry of renormalization.
Let $X= \mathbb{R}^\infty$. Then
the unordered configuration space of $n$ points in $\mathbb{R}^\infty$ is a model for the classifying space $B \Sigma(n)$ of the symmetric group $\Sigma(n)$;
the ordered configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-action, is a model for the $\Sigma(n)$-universal principal bundle.
Edward Fadell, Lee Neuwirth, Configuration spaces Math. Scand. 10 (1962) 111-118, MR141126, pdf
Craig Westerland, Configuration spaces in geometry and topology, 2011, pdf
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)
Edward R. Fadell, Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001), MR2002k:55038, xvi+313 pp.
F. R. Cohen, S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, MR2002m:55020