Let $C$ be a compact 2-dimensional cylinder $[0,1]\times S^1$. Let $A,A'$ be the two connected components of its boundary. 

Let $Q$ be a square. Let $a,a'$ be a pair of opposite edges of $Q$. 

Consider a smooth imbedding $f\colon Q\to C$ such that
$$f(a)\subset A,\, \, f(a')\subset A'.$$
Assume moreover that $f(\partial Q\backslash (a\cup a'))$ in contained in the interior of $C$.

**Question.** Is it true that $C\backslash f(Q)$ is connected?

A reference will be helpful.