There exist no continuous one one onto (Bijective ) function form $[0,1]$ $\to$ $[0,1]\times[0,1]$

Question

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 .

Answer 1

There is no such function because, since the domain is compact, it would be a homeomorphism. But $[0,1]$ and $[0,1]\times[0,1]$ are not homeomorphic: if you remove $\frac12$ from $[0,1]$, it becomes disconnected. But there is no such point in $[0,1]\times[0,1]$.