Configuration space of unordered points of a countable-dimensional real vector space
Let be a natural number.
Denote by a countable-dimensional real vector space over ; 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 s for initial coordinates.
We can now consider the corresponding configuration space of unordered points. If there are unordered points, this space, denoted , is the space of all possible configurations of distinct unordered points in .
The configuration space of 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 . Thus, this space can be viewed as a classifying space for the symmetric group of degree .