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We throw a fair coin countable but infinite times and $\Omega$ denotes the set of infinite sequences of either heads or tails. $\Omega$ is uncountable and I was curious whether one can show that using formal languages instead of an injective mapping $[0,1)\to\Omega$.

At first I was thinking about the alphabet $\Sigma=\{H,T\}$ and arguing about the properties of $\Sigma^*$ but I did not remember correctly that this does not include any infinite words over said alphabet. Is there still something that can be related to my example?

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Instead of $\Sigma^*$, the set you are looking for is $\Sigma^{\omega}$, which is the set of all infinite strings (more commonly known as streams). However, I don't see how this makes for a significantly different argument. In that case, you will have a set consisting of infinite strings of $H$'s and $T$'s, which is not conceptually different from having infinite strings of $0$'s and $1$'s, i.e. infinite bit strings, and then you are back to giving a mapping between infinite bit strings and real numbers between $0$ and $1$.

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