Q:

Smooth map $S^1 \to S^2$ can not be surjective

Why cannot a smooth (or piecewise linear) map $S^1 \to S^2$ be surjective?  There are space-filling curves, but the usual examples have very "twisty" definitions.
UPD A bit of background for this problem. It's part of he proof that all the normal vector fields on $S^1 \subset \mathbb{R}^4$ are homotopic. Once it's proved that every map $S^1 \to S^2$ is homotopic to piecewise linear or smooth that is what is left to prove the statement.
UPD One more thing. This is a first semester set of tasks. Sard's theorem is not exactly what gives intuition besides this problem. Given answer for PL is what I was looking for. ;) It would be great to find a reasoning like this for the smooth case.

A:

It may be easier for a PL map than a smooth one: the image of a PL curve lies in finitely many great circles. But no topological space is the union of finitely many nowhere dense sets (or for an unnecessarily big hammer use the Baire category theorem.)

