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?