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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 ?
This is the image you can have in mind when thinking about this solution. Other users have already given perfect explanations of solutions. I think this image will help with your intuition and aid you in wiring the proof yourself.
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 $f(x)=\tan\frac {\pi x}{2}$.
