TITLE: Is there a continuous function from the torus to the sphere? or from sphere to the torus?
QUESTION [6 upvotes]: I know that a sphere and a torus are not homeomorphic. But exists a continuous function from the torus to the sphere? or to be not homeomorphic implies that they are imposible (continuous function)

REPLY [7 votes]: Given two nonempty topological spaces, there is always a continuous map
from one to the other; simply map the first space to a point in the second space. So the question remains, are there any interesting maps, and of course that depends on what you find interesting.
From the sphere $S^2$ to the torus $T^2$ I'd say no. The second homotopy
group of the torus $\pi_2(T^2)$ is zero. So every continuous map from $S^2$
to $T^2$ is homotopic to a constant map.
From $T^2$ to $S^2$ I'd say yes. There's a double cover from
$T^2$ to $S^2$ (ramified at $4$ points; think elliptic curves). This induces a nontrivial map
between the second homology groups $H_2(T^2,\Bbb Z)$ and
$H_2(S^2,\Bbb Z)$ (both are isomorphic to $\Bbb Z$) so isn't homotopically trivial.