Are the sample space for one and multiple coin tosses the same? I am reading 'All of Statistics by Larry Wasserman'. I am at the first chapter and reading about sample space, sample outcome and events. I am a bit confused with one of the examples the author provided when explaining sample space.
Definition of sample space from the book: The sample space $\Omega$ is the set of possible outcomes of an experiment.
Here is what I understood: If our experiment is a coin toss, then the outcome of the experiment is either head or tail. So our sample space is $\Omega = \{H, T\}$ Also since sample space is a set it can not contain duplicate values.
Then the book goes on and gives us an example: If we toss a coin forever, then the sample space is the infinite set,
$\Omega = \{\omega = (\omega_1, \omega_2, \omega_3, . . . , ) : \omega_i ∈ {H, T}\}$
But this contradict with my understanding. Because in the example, both $\omega_1$ and $\omega_2$ can be H, and duplicate values are not allowed in a set.
My understanding is that, regardless of how many times we toss the coin(once or forever), the sample space will always contain two values($\Omega = \{H, T\}$) and not an infinite number of values, as the book says - because of the definition of set.
Can someone help me clear our my confusion.

Suppose we toss the coin three times.  Then there are $8$ different possible outcomes:
HHH, HHT, HTH, HTT, TTT, TTH, THT, THH
and the sample space is the set of these $8$ outcomes.  The generalization to infinite tosses should be clear.  The elements of the sample space are not simply "heads" and "tails" but sequences of heads and tails.
