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Consider a Coxeter group $(W,S)$ and a topological space $X$. We define a mirror structure on $X$ as a locally finite family $(X_{s})_{s\in S}$ of closed subspaces of $X$. Let's consider $W$ with the discrete topology. With this data we construct a space $\mathcal{U}(W,X)$ as the quotient space $\mathcal{U}(W,X):=(W\times X)/\sim$ where $\sim$ is the equivalence relation: $(w,x)\sim(w',y)$ if and only if $x=y$ and $w^{-1}w'\in W_x$, here $W_{x}$ denotes the subgroup of $W$ generated by the $s\in S$ such that $x\in X_{s}$. It's easy to see that there is an action of $W$ in $\mathcal{U}(W,X)$ given by $w'[w,x]=[w'w,x]$. It is also easy to see that the map $i:X\rightarrow\mathcal{U}(W,X)$ given by $i(x)=[1,x]$ is an embedding. Thus, we identify $X$ with it's image in $\mathcal{U}(W,X)$. We write $wX$ for the image of $\{w\}\times X$ in $\mathcal{U}(W,X)$. Is it true that there is a bijection between $W$ and the set $\{wX\}_{w\in W}$?

This is not homework, I just can't see if this is true or not.

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