The collision-free unordered configuration space of two planar points

Give ${\mathbb{R}}^4$ its Euclidean topology, whose basic open sets are the Euclidean open balls, and write a point as an ordered pair $(x,y)$ with $x,y\in {\mathbb{R}}^2$. The ordered configuration space is the subspace $F_2({\mathbb{R}}^2):=\{(x,y)\in {\mathbb{R}}^2\times {\mathbb{R}}^2:x\mathrm{\ne }y\}$ with the subspace topology. The transposition acts by $(x,y)\mapsto (y,x)$. The unordered configuration space of two distinct planar points is $C_2({\mathbb{R}}^2):=F_2({\mathbb{R}}^2)\mathrm{/}((x,y)\sim (y,x))$ with the quotient topology of the canonical projection $q:F_2({\mathbb{R}}^2)\to C_2({\mathbb{R}}^2)$. We write its points as unordered sets $\{x,y\}$; equality in this set means equality up to transposition. The diagonal $x=y$ is excluded, so collisions are not points of this space.

The topology on each finite Cartesian coordinate space used below is its Euclidean topology; on a finite product it is equivalently the topology with basis the products of open sets in the factors.

Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

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

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

This is a mathematical configuration space. Its definition alone makes no claim about a physical particle species, exchange statistics, or a realized holonomy.