Let $X$ be a sample space and define a probability distribution function $p:X\to \mathbb{R}$ such that $P(x)\geq 0$ and $\int p(x)dx=1$,let $S$ be a family of probability distributions on $X$ ,suppose that each element of $S$ ,may be parametrized using n real-valued variables $[\xi^1,....,\xi^n]$ so that
$$S=\{p_{\xi}=p(x;\xi):\xi=[\xi^1,...,\xi^n]\in E\}$$ where $E\subseteq \mathbb{R^n}$,we call such $S$ an n-dimensional statistical manifold.
let supp(p)=$\{x:p(x)>0\}$.Letting $X$ be redefined as supp(p),this is equivalent to assuming that $p(x;\xi)>0$ holds for all $\xi\in E$ and for all $x\in X$.This means that $S$ is a subset of $$P(X)=\{p:X\to\mathbb{R}:p(x)>0(\forall x \in X),\int p(x)dx=1\}$$
My question is how we are saying $S$ is a subset of $P(X)$. and what will be $S$ if we choose $X$ as supp(p)?
Also, the author says that when $X$ is a finite set we may consider $P(X)$ itself to be a statistical model which forms a $(|X|-1)$ dimensional manifold $(|X|$ denotes the cardinality of $X$).In this, we can consider $S$ is a submanifold of $P(X)$. How $S$ is a submanifold of $P(X)$ and what will happen if we have $X$ as an infinite set.