Configuration space of ordered points of a countable-dimensional real vector space

Definition

Let $n$ be a natural number.

Denote by $\mathbb {R} ^{\omega }$ a countable-dimensional real vector space over $\mathbb {R}$; explicitly, it is the space of sequences of real numbers with at most finitely many nonzero entries, under coordinate-wise addition and scalar multiplication. We put a topology on it using the coherent topology arising as the union of finite-dimensional $\mathbb {R} ^{k}$s for initial coordinates.

We can now consider the corresponding configuration space of ordered points. If there are $n$ ordered points, this space, denoted $F_{n}(\mathbb {R} ^{\omega })$, is the space of all possible configurations of $n$ distinct unordered points in $\mathbb {R} ^{\omega }$.

Facts

The configuration space of $n$ ordered points is a contractible space. The symmetric group of degree $n$ acts freely on this space by permuting the ordering of the points. The quotient space under this action is the configuration space of unordered points of a countable-dimensional real vector space, denoted $C_{n}(\mathbb {R} ^{\omega })$. That space is a geometric realization of the classifying space for the symmetric group of degree $n$.