This is an ideal case for using RL Moore’s way of defining connected sets.
Two non-empty sets are separated if neither contains a point or a limit point of the other.
Then A set is connected if and only if it is not the union of two separated sets.
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.
Then would I use that to assume to the contrary that a set is disconnected yet it is not the union of two non-empty disjoint open sets, so the disconnection must be of closed sets...? I am not sure how that definition follows to the proposition that I want to prove.
Edit: So here's my attempt at a proof:
Suppose are two non-empty disjoin open sets such that but suppose to the contrary that they do not form a disconnection of . Then, without loss of generality, . Since , it follows that there exists . So there exists such that and this ball contains elements in and since x is a boundary point of . This means there exists , which is a contradiction, so and since the index of the sets was arbitrarily fixed, it follows is in fact a disconnection.
But now I'm still having trouble proving the converse...