Unordered configuration spaces $C_n(X)$

Let $n\in \mathbb{N}$ and let $X$ be a topological space (Ordered configuration spaces $F_n(X)$, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). The label permutation action

$S_n\times F_n(X)⟶F_n(X),(\sigma ,x)⟼\sigma \cdot x,(\sigma \cdot x{)}_i=x_{{\sigma }^{-1}(i-1)+1},$

is a continuous free left action (The symmetric group acts continuously and freely on $F_n(X)$ by permuting labels). Its set of orbits $S_n\cdot x=\{\sigma \cdot x:\sigma \in S_n\}$ (The orbit $G\cdot x$ and stabilizer $G_x$ of a point in a group action) is the unordered configuration space of $n$ points in $X$, written

$C_n(X):=F_n(X)\mathrm{/}{S}_n=\{S_n\cdot x:x\in F_n(X)\},$

and it carries the quotient topology of the canonical projection

$p_n:F_n(X)⟶C_n(X),p_n(x):=S_n\cdot x,$

in the sense of The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection: a subset $V\subseteq C_n(X)$ is open exactly when $p_n^{-1}(V)$ is open in $F_n(X)$. This projection is a quotient map, hence continuous and surjective, and points of $C_n(X)$ are written $[x]:=p_n(x)$.

Basepoint. For a base configuration $q\in F_n(X)$ in the sense of Ordered configuration spaces $F_n(X)$, the basepoint of $C_n(X)$ is the orbit

$[q]=p_n(q)=S_n\cdot q,$

and $C_n(X)$ is nonempty exactly when $F_n(X)$ is, in which case a basepoint can be fixed. The unordered space is based by the orbit of the ordered base configuration, and this is the basepoint used in every later construction on this page.

Elementary cases. $C_0(X)$ is the quotient of the one-point space $F_0(X)$ by the trivial group $S_0$, hence is a one-point space. Since $S_1$ is trivial, $p_1:F_1(X)\to C_1(X)$ is a bijective quotient map and hence a homeomorphism: for every open $U\subseteq F_1(X)$, the equality $p_1^{-1}(p_1(U))=U$ makes $p_1(U)$ open by the quotient topology. Thus $C_1(X)$ is canonically homeomorphic to $X$ by $[(x)]\mapsto x$, using the single-coordinate homeomorphism $F_1(X)\to X$. These are canonical identifications, not literal equalities of the orbit set with the original set.

Elements are $n$-element subsets, as a set. Because the coordinates of a configuration are pairwise distinct, two ordered configurations $x,y\in F_n(X)$ lie in the same orbit exactly when their underlying sets of coordinates agree: if $y=\sigma \cdot x$ then the coordinates of $y$ are those of $x$ in a different order, and conversely, if $\{x_1,\dots ,x_n\}=\{y_1,\dots ,y_n\}$, then for every label $i$ there is exactly one label $\sigma (i)$ with $y_i=x_{\sigma (i)}$, and $i\mapsto \sigma (i)$ is a bijection of $\{1,\dots ,n\}$ (Injection, surjection, bijection); composing with the identification $\kappa (i)=i-1$ of labels with $n$, the permutation $\tau =\kappa \circ {\sigma }^{-1}\circ {\kappa }^{-1}\in S_n$ takes $x$ to $y$: $(\tau \cdot x{)}_i=x_{{\tau }^{-1}(i-1)+1}=x_{\sigma (i)}=y_i$. Hence

$C_n(X)⟶\{S\subseteq X:S\text{ has exactly }n\text{ elements}\},[x]⟼\{x_1,\dots ,x_n\},$

is a bijection of sets, the inverse sending an $n$-element subset $S$ to the orbit of any enumeration of $S$ (The cardinality $\mid A\mid$ of a finite set). This identifies the elements of $C_n(X)$ with $n$-element subsets of $X$; the topology on $C_n(X)$ is the quotient topology displayed above, and no topology on a set of subsets is asserted here. In particular the quotient topology is not defined through any metric or hyperspace structure.