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.