are the regular topology on $\mathbb R$ and the product topology on $\omega^\omega$ equivalent?

Question

are the regular topology on $\mathbb R$ and the product topology on $\omega^\omega$ equivalent?

By the product topology on $\omega^\omega$ I mean the topology in which an open basis set is a set of all sequence which has the same finite prefix.

If the answer is no, then, is one of them finer then the other?

Thank you!

Answer 1

Note that the topology on $\Bbb R$ is connected and locally compact, whereas the topology on $\omega^\omega$ is totally disconnected and every compact set has an empty interior. So the two of them are very different from one another.

For what it's worth, though, $\Bbb R$ can be obtained as a continuous image of $\omega^\omega$.