Why does a circle cut a torus into an annulus?

Question

Let $\phi : S^1 \rightarrow T^2$ be an (topological. Not necessarily smooth) imbedding of the circle in the 2-torus and let $\iota : S^1 \rightarrow T^2, \theta \mapsto (\theta,0)$ be the imbedding into the first factor. How can I show that there is a homeomorphism $\psi : T^2 \rightarrow T^2$ such that $\psi \circ \iota = \phi$? Another way to ask this is "why does $\phi$ cut the torus into an annulus?" Thank you for considering my question.