Ordered configuration spaces $F_n(X)$

Let $n\in \mathbb{N}$, so that $n=\{0,1,\dots ,n-1\}$ is the set of its predecessors (The natural numbers $\mathbb{N}$ (von Neumann)), and let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). Write

$X^n:={\prod }_{k<n}X$

for the $n$-fold product, carrying the product topology (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), and display its points as $(x_1,\dots ,x_n)$: the label $i\in \{1,\dots ,n\}$ names the coordinate of index $i-1$ in the sense of that definition. The ordered configuration space of $n$ points in $X$ is the subspace

$F_n(X):=\{(x_1,\dots ,x_n)\in X^n:x_i\mathrm{\ne }{x}_j\text{ whenever }i\mathrm{\ne }j\}$

with the subspace topology inherited from $X^n$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). Equivalently

$F_n(X)=X^n\setminus {\bigcup }_{i\mathrm{\ne }j}\{(x_1,\dots ,x_n)\in X^n:x_i=x_j\},$

since a tuple lies in $F_n(X)$ exactly when its entries are pairwise distinct: the collision diagonals $x_i=x_j$, $i\mathrm{\ne }j$, are removed from the product. Points of $F_n(X)$ are called ordered configurations of $n$ points in $X$.

The label set. The labels $1,\dots ,n$ are part of the data, and throughout this page they are identified with the set $n=\{0,1,\dots ,n-1\}$ by the bijection $\kappa (i):=i-1$. It is through $\kappa$ that the symmetric group $S_n=\mathrm{Sym}(n)$ acts on $F_n(X)$, in The symmetric group acts continuously and freely on $F_n(X)$ by permuting labels.

Elementary cases. For $n=0$ the product $X^0$ is a one-point space (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), and the defining condition is vacuous, so

$F_0(X)=\{\text{the empty tuple}\}$

for every $X$, including $X=\mathrm{\varnothing }$. For $n=1$ there is no pair $i\mathrm{\ne }j$, so single-coordinate evaluation $(x_1)\mapsto x_1$ is a canonical homeomorphism $F_1(X)\cong X$. For $n\ge 2$ and any $X$ one has

$F_n(X)\mathrm{\ne }\mathrm{\varnothing }⟺X\text{ has at least }n\text{ distinct points}\mathrm{.}$

if $X$ has at least $n$ points, an injection $\{1,\dots ,n\}\to X$ is exactly a tuple of pairwise distinct points of $X$, and conversely such a tuple displays $n$ distinct points. In particular $F_n(X)=\mathrm{\varnothing }$ when $X$ is empty and $n\ge 1$.

Based configurations. A base configuration in $F_n(X)$ is a point $q=(q_1,\dots ,q_n)$ of $F_n(X)$. Such a $q$ is fixed once and for all only when $F_n(X)$ is nonempty; when $n\ge 1$, $F_n(X)\mathrm{\ne }\mathrm{\varnothing }$, and $X$ is infinite, $F_n(X)$ is infinite: from any one configuration, keep coordinates $2,\dots ,n$ fixed and vary the first coordinate among the infinitely many points of $X\setminus \{q_2,\dots ,q_n\}$. The choice of $q$ is part of the data of every construction below. All base configurations on this page are chosen in the ordered space $F_n(X)$; the corresponding basepoint of the unordered quotient is its orbit (Unordered configuration spaces $C_n(X)$).

Separation of distinct coordinates. If $X$ is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and $q\in F_n(X)$, then the finitely many points $q_1,\dots ,q_n$ are pairwise distinct, and for each pair $i\mathrm{\ne }j$ Hausdorffness supplies disjoint open sets separating $q_i$ from $q_j$; a finite intersection over the finitely many pairs $j\mathrm{\ne }i$ therefore gives, for every $i$, an open neighbourhood $U_i$ of $q_i$ with $U_i\cap U_j=\mathrm{\varnothing }$ whenever $i\mathrm{\ne }j$. This is used in Disjoint coordinate neighbourhoods evenly cover the unordered configuration space.