I found the theorem and its proof in the attachment file.
The thing is that I don't understand why suddenly "either disjoint or equal" become "disjoint" for "Ix=Iy" condition.
Could you explain this for me?
I found the theorem and its proof in the attachment file.
The thing is that I don't understand why suddenly "either disjoint or equal" become "disjoint" for "Ix=Iy" condition.
Could you explain this for me?
It is actually easy. But you need to understand that is a connect set.
In the open connected sets are .
If two connected set have a point in common there union is connected and has one of those three forms.
Now by definition of is a maximum connected open set.
Thus, either .