I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like $\left(\Omega, \mathfrak{F}, \{\mathfrak{F}_n\}, \mathbb{P}\right)$ where $\mathfrak{F}_0\subset\mathfrak{F}_1\subset\cdots\subset\mathfrak{F}$. $n$ is the index indicating time. Let us keep ourselves to discrete time and sample space. This is just to make the notations clear.
If I consider tossing a coin giving head($H$) or tail($T$) nine times (completely random, just to keep it finite), then $\Omega=\{H, T\}^9$. Obviously $\mathfrak{F}=\mathfrak{F}_9=\mathcal{P}(\Omega)$ where $\mathcal{P}$ is the power set. But, how about $\mathfrak{F}_2$? I have the feeling that $\mathfrak{F}_2=\mathcal{P}\left(\{H, T\}^2\right)$, since that is the set of yes/no questions we can answer after two tosses. But can we really say that $\mathcal{P}\left(\{H, T\}\right)^2\subset\mathcal{P}\left(\{H, T\}^3\right)\cdots\subset\mathcal{P}(\Omega)$?
Can you connect the simple repeated toss experiment with the abstract definition of filtered space?