Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$.
How can I prove that S is homeomorphic to the sphere?
I'm studying an existence of triangulations proof and I'm stuck there. I need help. Thanks
Definition of finite character:
Let $\{U_i\}$ be a family of open sets in a surface S. We say $\{U_i\}$ is of finite character if the following conditions are satisfied:
(I) The family $\{U_i\}$ is locally finite, i.e, each point of S has a neighborhood which intersects only finitely many $U_i$'s
(II) The closure $\bar U_i$ of $U_i$ in S is a closed 2-cell
(III) Each $J:=∂\bar U_i$ meets at most finitely many other $J_j$'s
(IV) $J_i \cap J_j$ has finitely many connected components (which may be either arcs or points) for each i,j.