Why isn't $\displaystyle f: [0, 2 \pi ) \to S^1 $ a homeomorphism, where $\displaystyle f(x) = (\cos x, \sin x) $ and $\displaystyle S^1 $ is unit circle on the plane?
The book says that $\displaystyle S^1 $ is the unit circle on the plane.
Intuitively it says that you cant unwrap the unit circle onto the interval $\displaystyle [0, 2 \pi) $. But you can wrap the interval $\displaystyle [0, 2 \pi) $ onto the unit circle. Why is this?