How to show that the circle group T contains a copy of unit interval [0,1]?

Question

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?

Answer 1

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.

Answer 2

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