In the Wikipedia article about stochastic processes, a sample function is defined by specifying some $\omega\in\Omega$, as $X(\cdot,\omega):T\to S$, which then is a non-random function.
My (conceptual) question is: with $\{X(t,\cdot)\}_{t\in T}$ being a given family of random variables (i.e. a stochastic process), how would we get more than $|\Omega|$ different sample functions (which is something I intuitively expect to be the case)?
Let me illustrate this question with a Bernoulli process, e.g. a sequence of coin tosses. Let $X_i$ be the random variable that describes the outcome of the $i$-th toss. The $X_i$ are all independent, $\Omega =\{H, T\}$, and I may (may I?) specify that $X_i(H) = 1$ for all $i$, and $X_i(T) = 0$. In this case I would have exactly two choices for sample functions: Either $\omega = H$, giving me the sequence $[1,1,1,\ldots]$, or $\omega = T$, giving me $[0,0,\ldots]$. Intuitively, however, I would expect any binary sequence, e.g. $[1,0,0,1,0,\ldots]$, to be a possible sample function.
How can this be explained by picking a single $\omega$, and not a sequence of $\omega_i$? What am I missing in the definition?