How do I proof this is the definition of disconnectedness that I am using that is disconnected if there exists two sets such that they are both non-emtpy, and

Attempting to prove the first condition, that disconnected implies the open set union condition, I tried to show that if form a disconnection, then so does 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 . 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.