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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

user135541
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2 Answers2

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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!

Amitesh Datta
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  • could you please be more specific? I still don't get how to prove that T is a homeomorphism. thanks! – user135541 Apr 27 '14 at 04:26
  • Dear @user135541, do you know how to prove that $z\to \overline{z}+1+i$ is a homeomorphism $\mathbb{C}\to \mathbb{C}$? Do you know that, if $f:X\to Y$ is a homeomorphism (between topological spaces), then the restriction of $f$ to $A\subseteq X$ is a homeomorphism $A\to f(A)$? If you know how to prove both of these results, then you can prove that $T:\mathbb{R}\times [0,1]\to \mathbb{R}\times [0,1]$ is a homeomorphism. On which one (perhaps both) of these things are you stuck? – Amitesh Datta Apr 27 '14 at 05:44
  • A further hint: prove that $z\to \overline{z}-1+i$ is the inverse of $z\to \overline{z}+1+i$. – Amitesh Datta Apr 27 '14 at 05:48
  • I'm stuck on both. to show that a map is a homeomorphism, i have to show that the map is a continuous bijection. However, I cannot get an idea of how to prove this for the case $z \rightarrow \bar{z}+1+i$. I know the second result but I don't know how to prove it. – user135541 Apr 27 '14 at 05:49
  • Dear @user135541, to prove that a function is a bijection, you could construct an inverse (see my comment above). To prove that a function is continuous, you could use the $\epsilon$-$\delta$ definition of continuity: suppose I fix $\epsilon>0$. Can you find the $\delta>0$ such that $\left|z-w\right|<\delta$ $\implies$ $\left|(\overline{z}+1+i)-(\overline{w}+1+i)\right|<\epsilon$? (In fact, this proves the stronger (and true) claim that $z\to \overline{z}+1+i$ is not only continuous, but also uniformly continuous.) – Amitesh Datta Apr 27 '14 at 05:52
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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?

MPW
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