If I have a set $X=\{x_1,x_2,...,x_n:x_{i}\in\mathbb{R^3};x_{i}\neq x_{j} \text{ for }i\neq j\}$. and suppose I use the Gaussian mixture model to map fit the probability density function to this $X$ with $K$ components. The If I have $\mathbb{X}=\{X\}$ collection of all such $X$ and for every $X$ we fit the pdf so the set $S=\{p(x;\Theta)\}$, if I want to prove that $S$ is a statistical manifold then what should be the approach?\
I know that Give an $m$-dimensional topological manifold $\Theta$, and $1-1$ map from $\Theta$ to the space of probability density function $\theta\mapsto p(x;\theta)$ , the image of this mapping , denoted $S=\{p(x;\theta):\theta\in \Theta\} $is an $m-$dimensional statistical manifold.
$\textbf{We want the following:}$ we have to show that the space of parameters for GMM is topological space and the map from $\theta\mapsto p(x;\theta)$ is one to one.
can someone give some hint for this. Thanks