Configuration space of ordered points

Definition

Suppose $X$ is a topological space and $n$ is a natural number. The configuration space of ordered points $F_{n}(X)$ is defined as follows:

• As a set, it is the set of ordered $n$-tuples of distinct points of $X$.
• The topology is given as follows. First, note that $F_{n}(X)$ is the subset of $X^{n}$ comprising all $n$-tuples where no two coordinates are equal, i.e., it is the complement of the fat diagonal. We first give the product topology on $X^{n}$ and then give $F_{n}(X)$ the subspace topology arising from that.

Note that the term configuration space is typically used for the configuration space of unordered points $C_{n}(X)$, which is the quotient of this space under the equivalence relation induced by the action of the symmetric group $S_{n}$ by coordinate permutation.