It is known that no two distinct finite powers of the closed unit interval are homeomorphic:

$I^m$ is homeomorphic to $I^n$ iff $m=n$. (Brouwer, Lebesgue, 1911)

Is the analogous result for infinite powers of $I$ true?

That is, is it true that $I^\alpha$ is homeomorphic to $I^\beta$ iff $\alpha=\beta$, for cardinal numbers $\alpha$ and $\beta$?

(An affirmative answer would give a hope of defining a transfinite inductive dimension *for all spaces*, a problem essentially posed by Carl Menger.)

If not, what are the homeomorphism classes among the infinite powers of $I$?

Added Later: Since $|I^\alpha|=2^{\aleph _0 \alpha}=2^\alpha$ whenever $\alpha$ is infinite, assuming the Generalised Continuum Hypothesis, of course, $2^\alpha=2^\beta \Rightarrow \alpha=\beta$ and $I^\alpha$ and $I^\beta$ have different cardinalities if $\alpha$ and $\beta$ are distinct. But the answer below shows that they can be proved to be not homeomorphic within ZFC, and, more interestingly, even when they have the same cardinality (in some model which violates GCH).