Is there a continuous surjection from the closed unit square $[0,1]\times[0,1]$ to $\mathbb R ^2$?
If yes, please give examples. I'm a little stuck on this. What if I replace the closed unit square with the open one?
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No. The continuous images of unit square are compact so they are bounded in plane.
For the open case: take $f(x,y)=(\ln(-\ln x),\ln(-\ln y))$.