# How to prove that two sets are homeomorphic?

We have the interval $]-1,1[ \subset \mathbb{R}$. We have to say if this interval is homeomorphic with the open unit ball $U(0,1)$ of $\mathbb{R}^2$ and with the set $\mathbb{R}$.

I know that two sets are homeomorphic when it exists a continuous function between topological spaces that has a continuous inverse function. But how to prove that here ? I need an example to understand. For example, between $]-1,1[$ and $U(0,1)$. Someone could help me ?

• If you delete some element from $(-1,1)$ then the result is obiously not path-connected. If you do the same with unit ball $U(0,1)\subset\mathbb R^2$ then the result is path-connected. – drhab Jun 18 '17 at 13:17
• Why are you so sure this is true? Removing any point of $]-1,1[$ makes it disconnected, but the open unit ball $U(0,1)$ remains connected. – Demophilus Jun 18 '17 at 13:19
• No, I'm not sure ! We have to say IF the two sets are homeomorphic. Thank you, I believe that I have understood. :) – Mélanie De la Cheminée Jun 18 '17 at 13:23

We have the interval $]-1,1[ \subset \mathbb{R}$. We have to say if this interval is homeomorphic with the open unit ball $U(0,1)$ of $\mathbb{R}^2$ and with the set $\mathbb{R}$.
The unit ball case is already answered in comments. The standard homeomorhism $f$ between the interval $]-1,1[$ and $\mathbb{R}$ is given by the formula $f(x)=\tan\frac {\pi x}{2}$.