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 defining ingredient::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 defining ingredient::subspace topology arising from the defining ingredient::product topology on $$X^n$$. Then, we give $$C_n(X)$$ the defining ingredient::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.