Let $I=[-1,1]$ and $\operatorname{Hom}(I^n)$ be the set of topological homeomorphisms between $I^n$ and itself. Is there a classification of this set? What is the size of this set? What about $\operatorname{Diff}(I^n)$ (replace homeomorphisms with diffeomorphisms)?
Thanks!
On one hand, any homeomorphism is continuous, so it is determined by its values on the (countable) dense set of points with all coordinates rational. Therefore there are at most $|I^n|^{\aleph_0}=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}$ such homeomorphisms.
On the other, we can find $2^{\aleph_0}$ distinct homeomorphisms of $I^n$. The simplest way to see this is to note than if $f:I\to I$ is a homeomorphism, then $(x_1,x_2,\dots,x_n)\mapsto(f(x_1),x_2,\dots,x_n):I^n\to I^n$ is a homeomorphism as well, so we just have to find continuum many self-homeomorphisms of $I$. One such family is given by quadratic functions: consider $f_c(x)=x+c(x-1)(x+1)$ for $-\frac{1}{2}<c<\frac{1}{2}$.
Therefore $|\mathrm{Hom}(I^n)|=2^{\aleph_0}$. The exact same reasoning shows $|\mathrm{Diff}(I^n)|=2^{\aleph_0}$.