Does this space exist? Does the space $[0,1]^r$ exist with $r\in\mathbb R$?
And the Hausdorff dimension is $r$
Thank you very much

Not directly:
In the notation $[0,1]^n$ with $n\in\mathbb{N}$, we mean $[0,1]\times[0,1]\times\cdots\times[0,1]$, i.e. the $n$-times cartesian product.
There is no $r$-times cartesian product for $r\notin\mathbb{N}$, because you cannot repeat an operation (taking cartesian product) for a non-natural number amount of times.
Of course, you could construct a space with hausdorff dimension $r$ for any $r\in\mathbb{R}_{>0}$, but this would not be related directly to the notation $[0,1]^n$ for $n\in\mathbb{N}$, and there is probably more than one way to do it.
One such construction would be to make $[0,1]^r = [0,1]^{\lfloor r\rfloor}\times S_{\mathrm{frac}(r)}$ where $\mathrm{frac}(r)=r-\lfloor r\rfloor$ is the fractional part of $r$ and $S_{\mathrm{frac}(r)}$ is your favorite set of Hausdorff dimension $\mathrm{frac}(r)$, e.g. some generalised Cantor set.
