Is there a difference between $\mathbb R^\infty$, $\mathbb{R^N}$, and $\mathbb R^\omega$? I've seen these three notations and I'd like to know the difference:

$\mathbb{R}^\infty$
$\mathbb{R}^\mathbb{N}$
$\mathbb{R}^\omega$
$\mathbb{R}^{\mathbb{N}}$ and $\mathbb{R}^{\omega}$ are almost always synonyms, and both refer to the set of functions $\mathbb{N}\to\mathbb{R}$ (aka sequences of real numbers indexed by $\mathbb{N}$).  The symbol $\omega$ is an alternate name for the set of natural numbers (or more specifically, for one particular set-theoretic construction of the natural numbers) which is commonly used in set theory.  Conceivably someone might distinguish $\mathbb{R}^\mathbb{N}$ and $\mathbb{R}^\omega$ as, for instance, $\omega$ is always taken to include $0$ but some authors exclude $0$ from $\mathbb{N}$.  I've never seen anyone actually intentionally make such a distinction, though.
$\mathbb{R}^\infty$ is sometimes a synonym for $\mathbb{R}^\mathbb{N}$, but sometimes it instead refers to the subset of $\mathbb{R}^\mathbb{N}$ consisting of those functions $\mathbb{N}\to\mathbb{R}$ that are $0$ on all but finitely many inputs.
Hints : A basic open set of $\mathbb{R}^{\omega}$ is of the form
$$
U = U_1\times U_2\times \cdots \times U_n \times \mathbb{R} \times \mathbb{R} \times \cdots
$$
where $U_i$ are open sets in $\mathbb{R}$
similarly  for $\mathbb{R}^\infty$ and  $\mathbb{R}^\mathbb{N}$