Characterize the set $E$ such that $E$ and $E^\mathbb{N}$ are equipotent

Question

Can we find some nice characterizations for the set $E$ such that $E$ and $E^\mathbb{N}$ are equipotent? For example, we know that is true when $E = \mathbb{R}$, but it's false when $E$ is finite set or $E = \mathbb{N}$. A possible characterization is that: $E \sim E^\mathbb{N}$ $\iff$ there exists a set $T$ such that $T^\mathbb{N} \sim E$.
What else?

Answer 1

There is no good, and easy way to characterize this, because cardinal arithmetic is pretty wonky when all you assume is $\sf ZFC$ (which you don't even have if you work in a naive set theoretic setting).

The problem is that there are arbitrarily large sets $E$ such that $E^{\Bbb N}$ is strictly larger than $E$ itself. This is a consequence of König's theorem. And moreover, we don't really have provable bounds or control on what would the cardinalities of these $E^{\Bbb N}$ going to be. In principle, it could be the least possible cardinal, or it could be much larger, and so if there are cardinals in that gap, they will have to jump up when taking exponents by $\Bbb N$, and if not, then they don't have to.

Ultimately, $E^\Bbb N\sim E$ if and only if $E^\Bbb N\sim E$. And there isn't that much more you can say in the general case.