elementary set theory - 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.