Homeomorphism between the open sets of the circle and the open sets of real line

[ravikrocha]

I'm trying to prove the homeomorphism between the open intervals
of the real line and the open sets
of the circle with the induced topology of R^2.
Notice that the open sets of the circle is the intersection between
the open balls in R^2 and the circle itself.
Anyone can help me?

thank you.

 

[jbunniii]

I'm trying to prove the homeomorphism between the open intervals
of the real line and the open sets
of the circle with the induced topology of R^2.
Notice that the open sets of the circle is the intersection between
the open balls in R^2 and the circle itself.
Anyone can help me?

thank you.

Can you name any bijections (not necessarily homeomorphisms) between $\mathbb{R}$ and the circle? Then we can check whether these are homeomorphisms, or if there's an obvious modification to make them so.