homeomorphism $T: X \rightarrow X$

Question

How do i prove that $Tz=\bar{z}+1+i$ defines a homeomorphism $T: X \rightarrow X$ where $X=\mathbb{R}\times[0,1] \subset \mathbb{C}$ ? (how can there be a continuous bijection in this case?)

Also, how do I show that if G is the group of homeomorphisms generated by T, then X/G is the Mobius strip?

Any help would be greatly appreciated. thanks

Answer 1

Hints:

$z\to \overline{z}+1+i$ is a homeomorphism $\mathbb{C}\to \mathbb{C}$

The image under this homeomorphism of $\mathbb{R}\times [0,1]$ is $\mathbb{R}\times [0,1]$.

What is the orbit of $z\in \mathbb{R}\times [0,1]\subseteq \mathbb{C}$ under the action of $G$?

Hope this helps!

Answer 2

Hint: $T$ is the composition of a reflection and a translation. Can you show that each of these is a homeomorphism of its domain with its image?