Let $(\Omega, \mathcal F, P)$ be a probability space, and let $(S, \mathcal S)$ a set $S$ together with a $\sigma$-Algebra over $S$, also let $T$ be some index set, then for each $t \in T$ let $X_t : \Omega \to S$ be a random variable (i.e. a measureable function form $\Omega$ to $S$). Then the set $\{ X_t : t \in T \}$ is called a stochastic process.
Example: Let $T = \mathbb N_0$ and $\Omega = \{ H, T \}^{\mathbb N}$, then define $$ X_t(\omega) := \left\{ \begin{array}{ll} 1 & \mbox{if } \omega = H \\ 0 & \mbox{else } \omega = T \end{array}\right. $$ for $t > 0$ and $X_0(\omega) := 0$ and $Z_t(\omega) := Z_{t-1}(\omega) + X_t(\omega)$ with $Z_0(\omega) := 0$. This process models an infinitely often, but at discrete moments, tossed coin, and $\{ Z_t \}_{t\in T}$ counts how often came head at the $t$-th toss.
By considering the above example I came to the suspection, that $\Omega$ must always be a cartesian product over $T$ as index set, and the $X_t$ relate in some way to the projections. Is this true, is there any reasonable stochastic process where $\Omega$ is not an cartesian product over $T$? And also for some $(\Omega, \mathcal F, P)$ probability space and index set $T$, then the product probability space $(\Omega^T, \mathcal F', \mathcal P')$ where $\Omega_t = \Omega$ over $T$ with $\pi_t(\omega') = \omega'(t)$ are the projection maps, then $\{ \pi_t : t \in T \}$ is some simple (or trivial) example of a stochastic process?
The $d$-dimensional Brownian motion is often defined on $\Omega=C^0([0,+\infty),\mathbb R^d)$, which is not a product space. Then $X_t(\omega)=\omega(t)=\pi_t(\omega)$ for every $t$ and every $\omega$ in $\Omega$ hence, indeed, $(X_t)=(\pi_t)$.