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Here, $T$ is the set of all complex numbers of absolute value 1. I want to show that there is a (natural) copy of the interval $[0,1]$. Any hint?

user73564
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  • You should say explicitly what you mean by a copy of the interval: do you mean an injection, and embedding as a set, a topological space, a manifold? – tomasz Apr 20 '13 at 23:09

2 Answers2

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$\phi: [0, 1] \to T$, $\phi(t) = e^{i \pi t}$. This map is a homeomorphism between $[0, 1]$ and upper semi-circle.

xyzzyz
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Consider the map $\varphi : [0,1] \to T$ defined by $\varphi(t) = e^{\pi i t}$. I believe it has all the desired properties, such as continuous and is a homomorphism.

Suugaku
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