Homeomorphism

[Siron]

Hi,

How can I prove there's no homeomorphism between $\mathbb{R}$ and $\mathbb{R}^2$. I thought they are topological almost the same.

Thanks in advance.

 

[Deveno]

they almost are, there's just "one point of difference".

what happens to the connectedness of $\Bbb R$ when you remove one point? is this same true for $\Bbb R^2$?

(surely if they were homeomorphic this wouldn't happen, right?)

 

[Siron]

they almost are, there's just "one point of difference".

what happens to the connectedness of $\Bbb R$ when you remove one point? is this same true for $\Bbb R^2$?

(surely if they were homeomorphic this wouldn't happen, right?)

I guess if I remove one point $x \in \mathbb{R}$ then $\mathbb{R}$ is not connected anymore because it wouldn't be an interval then. An interval $X$ is defined as $\forall x,y \in X: x \leq y \leq z \Rightarrow y \in X$, which is not true in the case of $\mathbb{R}\setminus \{0\}$ for example, because if we take $x=-1, y=0 $ and $z=1$ then if $\mathbb{R}$ would be an interval we would have $0 \in \mathbb{R}$ which is not true.

Thus I guess that $\mathbb{R}^2$ is still connected if we remove one point. But how do I prove that? Maybe with path connectedness?