Let $\Sigma$ be a simply connected surface possibly non-compact with boundary. Is it true that any simple closed curve in $\Sigma$ bounds a disc?
When $\Sigma=\Bbb R^2,\Bbb S^2$ this is certainly true by the Jordan-Brouwer Separation Theorem. Any reference or proof will be appreciated. Thanks in advance.
Edit: If $\Sigma$ is a simply-connected non-compact surface with boundary and the simple closed curve is contained in $\text{Int}(\Sigma)$, then $\text{Int}(\Sigma)$ is homotopically equivalent to $\Sigma$ by collar-neighborhood theorem, and so $\text{Int}(\Sigma)$ is simply connected. Also, $\text{Int}(\Sigma)$ is non-compact, hence $\text{Int}(\Sigma)=\Bbb R^2$ by Uniformazation theorem. So, we are done Jordan curve theorem.