What are some of the consequences of Cantor's famous "I see it, but don't believe it" result showing a bijective mapping from $[0,1]$ to $[0,1]^n$? Cantor wrote in a letter to Dedekind "I see it, but don't believe it!" While discussing his discovery of a bijection between the interval $I=[0,1]$ and $I^n$.  While this is certainly a neat result, I can't think of where this becomes a wholly useful fact.  In what mathematical context has Cantor's result proven useful?  Are there are any notable cases where Cantor's discovery was utilized in a proof?  Are there any particular math problems which require knowing $|I|=|I^n|$ to solve?
Sorry if this is question is vague.  I recently read an article on Cantor's letter to Dedekind but cannot find any specific examples of his discovery being cited.

It's not a useful fact. 
It is an important fact; it shows that we can't distinguish $[0, 1]$ from $[0, 1]^n$ just by their cardinality, so if we want to make precise the intuition that they are really different (e.g. because the former has dimension $1$ and the latter has dimension $n$) we need to do something else (e.g. invent topological dimension, which involves inventing topology). 
