Find an equivalence relation ~ on [0,1] such that [0,1]/~ is homeomorphic to [0,1] x [0,1].

Question

I've thought a lot about this question and it doesn't seem possible to me. I'm assuming there is some special trick that I am not aware of. Any help is appreciated.

Answer 1

Steve D has given 99% of the answer. It is known that there exists a continuous surjection $p : I \to I^2 = I \times I$. Since domain and range are compact, $p$ is an identification map. Now define $s \sim t$ if $p(s) = p(t)$. This is an equivalence relation. Let $\pi : I \to Q = I/ \sim$ denote the quotient map. Then $p$ induces a unique function $\hat{p} : Q \to I^2$ such that $\hat{p} \circ \pi = p$. It is bijective by construction and continuous by the universal property of the quotient. The function $\hat{p}^{-1}$ satisfies $\hat{p}^{-1} \circ p = \pi$ and is therefore continuous because $p$ is an identification.