gn.general topology - how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B subset mathbb{R}^k$:

$dim(B) = min{n in mathbb{N}: exists K subset mathbb{R}^n, ~{rm s.t.} ~ B sim K}$,

where $sim$ denotes "homeomorphic to". Obviously, $0 leq dim(B) leq k$.

I have three questions: Given a $B subset mathbb{R}$,
1) As $k to infty$, how slow can $dim(B^k)$ grow? Can we choose some $B$ such that $dim(B^k) = o(k)$ or even $O(1)$?
2) Will it make a difference if we drop the Borel measurability of $B$ or add the condition that $B$ has positive Lebesgue measure?
3) Does this dimension-like notion have a name? The dimension concepts I usually see are Lebesgue's covering dimension, inductive dimension, Hausdorff dimension, Minkowski dimension, etc. I do not think the quantity defined above coincides with any of these, but of course bounds exist.

Thanks!