Does there exist a continous surjective map between $\mathbb R^2\to S^1$

Question

Does there exist a continuous surjective map between $\mathbb R^2\to S^1$?

I had find such map with domain $\mathbb R^2/${$0$}.

But I do not able to find if we insert 0 there?

Any help will be appreciated

Answer 1

How about $f(x,y)=(\cos x,\sin x)$?

Answer 2

Take $\pi:\mathbb{R}^2\to \mathbb{R}$ given by $\pi(x,y)=x$, followed by $\phi:\mathbb{R}\to S^1$ given by $\phi(t)=e^{2\pi i t}$. Both of these maps are continuous and it's easy to see that their composition yields a surjection. So, $\phi\circ \pi:\mathbb{R}^2\to S^1$ is a continuous surjection.

Answer 3

R^2 to R projection and then from R to circle the exponential map.