Let $\sim$ be an equivalence relation on the permutations of $(1,...,n)$. I thought about the topological spaces $\Bbb R^n/\!\sim$, i.e. $\Bbb R^n$ with standard topology after associating certain permutations $\sigma$ of the indices. More precisely:
$$(x_1,...,x_n)\sim_{\Bbb R^n}(x_{\sigma(1)},...,x_{\sigma(n)})\quad\Longleftrightarrow\quad (1,...,n)\sim(\sigma(1),...,\sigma(n)).$$
Example: $\Bbb R^n/S_n:=\Bbb R^n/\!\sim$ with $\sim$ associating all permutations, is the space in which the order of the coordinates does not matter.
I made the following observations:
- $\Bbb R^n/S_n\cong \Bbb R^n_+=\Bbb R^{n-1}\times\{x\in\Bbb R\mid x\ge 0\}$, the $n$-dimensional half-space. This can be seen by associating $\Bbb R^n/S_n$ with the subset $\{(x_1,...,x_n)\mid x_1\le\cdots\le x_n\}\subset\Bbb R^n$ with increasing coordinates.
- $\Bbb C^n/S_n\cong \Bbb C^n$ (a homeomorphism is given by Vièta's map which associates unordered roots with ordered coefficients of polynomials). This can be expressed as $\Bbb R^{2n}/\!\sim\;\cong \Bbb R^{2n}$ where $\sim$ only associates permutations in which certain consecutive pairs of indices permute in lockstep, e.g.
$$(\color{red}{1,2},\color{blue}{3,4})\sim(\color{blue}{3,4},\color{red}{1,2})\qquad\text{but not}\qquad (1,2,\color{red}3,\color{blue}4)\sim(1,2,\color{blue}3,\color{red}4).$$
Question: I was wondering what different spaces can emerge from $\Bbb R^n/\!\sim$. I would conjecture that it only can be $\Bbb R^n$ and $\Bbb R^n_+$. In this case I was wondering whether there is a simple characterization of the relations $\sim$ which generate either one.