What is $\mathbb R^\omega$?

Question

I have seen $\mathbb R^\omega$ mentioned in my topology texts but cannot find where $\omega$ is defined. Could someone please tell me what it means in comparison to $\mathbb R^n$?

Answer 1

The cardinality of $\Bbb N$, which is usually written $\aleph_0$ (the smallest infinite cardinal), is sometimes/often also written $\omega$. In other words

$$|\Bbb N| = \aleph_0 = \omega$$

Hence $\Bbb R^{\omega} = \Bbb R \times \Bbb R \times \Bbb R \times \dots$ is the cartesian product of countably infinitely many $\Bbb R$.