Function that wraps unit circle twice around itself I'm looking for an example of a function $f: S^1 \to S^1$ (from unit circle to unit circle), that is continuous, open and surjective but not injective. I have intuitively thought of example, that would be a function that "wraps" unit circle twice around itself. More precisely this is what I had in mind. We can uniquely write points on unit circle as $(\cos t, \sin t)$, where $t \in [0, 2 \pi)$. And the function would be $f: S^1 \to S^1$ with $(\cos t, \sin t) \to (\cos 2t, \sin 2t)$.  Now, I'm not even sure if this would be a good thing to look at, and if it was, I don't know how to show that $f$ is continuous and open.
Keep in mind that this is for course of real analysis and I'm not allowed to use facts and parametrizations from complex analysis.
Any comments or suggestions are very welcomed.

To formally define the map you describe consider the map $t\mapsto 2t\mapsto (\cos(2t),\sin(2t))$. Clearly, this is a continuous map $\phi\colon [0,2\pi]\to S^1$ as it is a composition of continuous maps. It is,

*

*Open, since it is a composition of open maps.

*Surjective, since it is a composition of surjective maps.

*Not injective, as it is two-to-one.

The map $\phi$ descends to the quotient of $[0,2\pi]$ identifying the endpoints giving a map $\widetilde{\phi}\colon S^1\to S^1$. This is certainly surjective, and the properties of being open and continuous also decends to the quotient by properties of the quotient topology (look at the properties characterizing the quotient topology and chase around the commutative diagram one obtains).
