As explained in this answer, it is possible to create a bijection from $[0,1]\rightarrow[0,1]^2$. However, the example provided is clearly not continuous. It seems either very complicated or impossible to create a continuous bijection between the unit interval and the unit square. Does this mapping exist? If so, what does it look like? If not, how does one prove that there does not exist a continuous bijective mapping between $[0,1]\rightarrow[0,1]^2$?
[Math] Continuous mapping from $[0,1]$ to $[0,1]^2$
continuityelementary-set-theorygeneral-topology
Best Answer
No such bijection exists: it is a standard result that any continuous bijection from a compact space (such as $[0,1]$) onto a Hausdorff space is a homeomorphism, and there's no homeomorphism from the unit interval to the unit square, since removing an interior point will make the first one disconnected, while leaving the latter connected.