1. ## circle heomomorphism

Why isn't $f: [0, 2 \pi ) \to S^1$ a homeomorphism, where $f(x) = (\cos x, \sin x)$ and $S^1$ is unit circle on the plane?

2. What is $S^1$?

3. The book says that $S^1$ is the unit circle on the plane.

Intuitively it says that you cant unwrap the unit circle onto the interval $[0, 2 \pi)$. But you can wrap the interval $[0, 2 \pi)$ onto the unit circle. Why is this?

4. You do realize that for $f$ to be a homeomorphic it must be bicontinuous?
That is, both $f$ and $\overleftarrow f$ must be continuous.

5. Yes I do. Then $\overleftarrow{f}$ is not continuous?

6. What would $\overleftarrow f$ even be?

7. Wait, I think $\overleftarrow{f}$ is not continuous. Thats why. Something about how the points dont converge to 0 but $2 \pi$ instead.