Is the closure of a set in a surface homeomorphic to the closure of its image under a chart?

Question

Consider a surface $\mathfrak S$ and an open set $A \in \mathfrak S$. If $A$ is homeomorphic to an open disc and $\partial A$ is homeomorphic to a circle, does this mean $\overline A$ is necessarily homeomorphic to a closed disc?

Answer 1

Not if $\mathfrak S$ is allowed to have a boundary: Consider $$\mathfrak S=\{\,(x,y)\in\Bbb R^2\mid x^2+y^2<4\land (y\ne 0\lor x\ge -1)\,\}$$ and $$A=\{\,(x,y)\in \mathfrak S\mid x^2+y^2>1\,\}.$$