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

Definition
Let $$n$$ be a natural number.

Denote by $$\R^\omega$$ a defining ingredient::countable-dimensional real vector space over $$\R$$; 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 $$\R^k$$s for initial coordinates.

We can now consider the corresponding defining ingredient::configuration space of unordered points. If there are $$n$$ unordered points, this space, denoted $$C_n(\R^\omega)$$, is the space of all possible configurations of $$n$$ distinct unordered points in $$\R^\omega$$.

Facts
The configuration space of $$n$$ unordered points is a path-connected aspherical space and its universal covering space is the corresponding configuration space of ordered points. The fundamental group is the symmetric group of degree $$n$$. Thus, this space can be viewed as a classifying space for the symmetric group of degree $$n$$.