Configuration space of ordered points of a countable-dimensional real vector space
Definition
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 ordered points. If there are ordered points, this space, denoted , is the space of all possible configurations of distinct unordered points in .
Facts
The configuration space of ordered points is a contractible space. The symmetric group of degree 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 . That space is a geometric realization of the classifying space for the symmetric group of degree .