Configuration space of unordered points

Definition

Suppose $X$ is a topological space and $n$ is a natural number. The configuration space of unordered points $C_{n} (X)$ (often simply called the configuration space), sometimes also denoted $(\binom{X}{n})$ , is defined as follows:

As a set, it is the set of $n$ -element subsets of $X$ .
The topology is given as follows: A $n$ -element subset of $X$ can be thought of as an orbit under the action of the symmetric group $S_{n}$ on the configuration space of ordered points $F_{n} (X)$ (defined as the subspace of $X^{n}$ comprising points which have pairwise distinct points). In other words, as a set $C_{n} (X) = F_{n} (X) / S_{n}$ . We give this a topology as follows: first, we give $F_{n} (X)$ the subspace topology arising from the product topology on $X^{n}$ . Then, we give $C_{n} (X)$ the quotient topology under the equivalence relation induced by the action of $S_{n}$ .

Facts

The configuration space of unordered points is not a homotopy invariant. In other words, if $X$ and $Y$ are homotopy-equivalent spaces, it does not necessarily follow that $C_{n} (X)$ and $C_{n} (Y)$ are homotopy-equivalent.