Union of 2 non-disjoint open intervals is open interval. I will not use word open later.
I want to obtain formal proof of this fact that is different from mine.
We know that union of two open sets is open set. Assume it is not an interval. Then it must be a union of intervals (because any open set is union of intervals). But union of more than one interval is not connected - we can just apply defintion of connectedness. And we know that union of nondisjoint connected sets is connected. Contradiction shows it must be an interval.
I am very unhappy with my attemp because I am trying to prove that any open set is union of disjoint intervals and in order to do I must prove that that union of non-dijsoint open intervals is open interval. So it is just cheating from my side, and in fact I do not understand two proofs instead of one proof. In fact I am not even sure that my attemp could be saved as a proof.
Or maybe there is another way to deal with these 2 propositions at once without logic flaws?