TITLE: Does there exist a continuous onto function from $S^2$ to $S^1$?
QUESTION [8 upvotes]: Here $S^2$ is 2 dimensional unit sphere in $\mathbb{R^3}$ and $S^1$ is unit circle in $\mathbb{R^2}$. Does there exist a continuous onto function from $S^2$ to $S^1$? Till now, I am able to construct a continuous map from $S^2$ to closed unit disk in $\mathbb{R^2}$( from projection map from $\mathbb{R^3}$ to $\mathbb{R^2}$). Continuous map from closed unit disc to unit circle is defined in the following way: First we construct an onto map $f$ from closed unit disc to $\mathbb{R}\cup \{\infty\}$. $f(x,y)=\frac{1}{\sqrt{x^2+y^2}-\frac{1}{2}}$ if $\sqrt{x^2+y^2}\neq\frac{1}{2}$ and $f(x,y)=\infty$ if $\sqrt{x^2+y^2}=\frac{1}{2}$. As we know that $\mathbb{R}\cup \{\infty\}$ is homeomorphic to unit circle  we can construct an onto continuous function from closed unit disc to unit circle using $f$. The composition of function from $S^2$ onto unit disc, and function from unit disc to circle is continuous onto function. Is my reasoning correct? If not, can one construct an onto continuous function from $S^2$ to $S^1$?

REPLY [2 votes]: Since each map is surjective the composition is also a surjective map