Proof that the interval and the plane is not homeomorphic I tried to prove this by contradiction. I assumed that there is a homeomorphism $f: (0,1) \longrightarrow \mathbb{R^2}$. A submap $ab(f): (0,1)\setminus\{\frac{1}{2}\} \longrightarrow \mathbb{R^2}\setminus\{f(\frac{1}{2})\} $ (it's also homeomorphism). It is obvious that the first space is not connected space, but I very confused about $\mathbb{R^2}\setminus\{f(\frac{1}{2})\}$. As far as I understand, it's something like a plane with a hole, but I can't prove that this space is connected.
I would be pleased if you help me with proof of this fact or if you give me some hints about my original question.

Well you can show, that there is always a path connecting two points. 
