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

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Definition

Let n be a natural number.

Denote by \R^\omega a 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^ks for initial coordinates.

We can now consider the corresponding 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.