Let $M^2$ be a smooth $2$-dimensional and simply connected manifold without boundary. Let $\Omega \subseteq M$ be an open and path-connected subset of $M$. Let us assume that each connected component of $\partial \Omega$ is contractible. Is $\Omega$ simply connected? I believe the answer is yes, but I haven't thought about those topological questions in a very long time. Any help will be very much apreciated.
EDIT: My intuition is that if by contradiction $\Omega$ is not simply connected, then I can find a loop in $\Omega$ wich is not homotopically trivial (in $\Omega$) but it is trivial in $M$. Somehow, for this to happen, there must be a loop in $\partial \Omega$, but this is not the case, by hypotesis.
SECOND EDIT As pointed out in the comments, the answer might be "no" if I don't assume something more on $\partial \Omega$. Actually, in the case I am interested in, $\partial \Omega$ is a union of at most countably many properly embedded 1-dim manifolds.