# Is there a continuous surjection from the closed unit square $[0,1]\times[0,1]$ to $\mathbb R ^2$?

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?

• For the second part, do you know an answer for an open interval and the real line? – Mark Bennet Sep 9 '14 at 17:00
• Please make the body of your question self-contained. The title is important, but should be separate from the body. – Asaf Karagila Sep 9 '14 at 17:02

• @you-sir the closed unit square is compact. The open unit square is in fact homeomorphic to the plane (hint: $\tan$). – Najib Idrissi Sep 9 '14 at 19:55
For the open case: take $f(x,y)=(\ln(-\ln x),\ln(-\ln y))$.