Precise meaning of $\omega$ as an ordinal number

Question

Let $2^\omega$ be the Cantor space.

I know that $\omega$ denotes the first transfinite ordinal, i.e. the order-type of the natural numbers. I am however still a bit confused about what that means exactly. Does that mean that $2^\omega$ is the cartesian product of denumerably many copies of $2$ or countably (i.e. finitely or denumerably) many copies of $2$?

Answer 1

The same as $2^N,$ denumeral (infinitely countable) many
copies of 2. This is not Cantor's set. It is a space
homeomorphic to the Cantor set.