Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\mathbb{R})$ is in the image of the map on $\Delta_n \times \mathbb{R}^n$, where $\Delta_n$ is interior of the $n$-simplex (i.e.: $k_1,\dots,k_n \in (0,1)$ with $\sum_{i=1}^n k_i =1$, taking $(k_1,\dots,k_n)\times (x_1,\dots,x_n)$ to $\sum_{i=1}^n k_j \delta_{x_i}$. Clearly this map is continuous, when $\mathcal{P}_{n:+}(\mathbb{R})$ is equipped with the Prokhorov metric.

However, is it a *covering map*? I have not been able to disprove it so I'm thinking maybe it is...?