How do I proof this is the definition of disconnectedness that I am using that $\displaystyle S$ is disconnected if there exists two sets $\displaystyle S_{1}, S_{2}$ such that they are both non-emtpy, $\displaystyle S_{1} \cup S_{2} = S$ and $\displaystyle cl(S_{1})\cap S_{2} = cl(S_{2})\cap S_{1} = \emptyset$

Attempting to prove the first condition, that disconnected implies the open set union condition, I tried to show that if $\displaystyle (S_{1},S_{2})$ form a disconnection, then so does $\displaystyle (S_{1} - \partial S_{1}, S_{2}-\partial S_{2})$ as well, but I am not even sure if this is the correct approach because I am not nsure how to show that those two sets would be non-empty and that their union would be $\displaystyle S$. And I am not sure at all how to show the other direction of the proof (that the open set union condition implies disconnectedness). Any help would be appreciated.