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.