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I wanted to prove that there exist no continuous one to one and onto function form $[0,1] \to [0,1]\times[0,1]$.
My attempt : image of f on $[0,1]$ , a compact set is again compact .$[0,1]\times[0,1]$ is also compact.
On contrary suppose there exist continuous one one onto function between $[0,1]\to [0,1]\times[0,1]$ then its inverse function is also continuous one one onto.
Upto this I can write from given information
No idea how to proceed further .Any Help will be appreciated .