Consider an example of flipping a coin infinitely many times. In class, the following notation was used:
$$\Omega=\{H,T\}^{\mathbb{N}}$$
Although I understand what $\Omega$ is supposed to look like, (infinite numerations of the infinite combinations of Heads and Tails), what is the sense/logic behind this notation?
For sets $A$ and $B$, the notation $B^A$ is used for the set of functions from $A$ to $B$. (Possible reason for this is that if $A$ and $B$ are finite, then $\mid B^A \mid = \mid B \mid^{\mid A \mid}$, where $|S|$ is the number of elements in set $S$).
A sequence of coin tosses can be regarded as a function from $\mathbb{N}$ to $\{H,T\}$ where $1\mapsto$ first toss, etc.