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 $\displaystyle I_x$ is a connect set.
In $\displaystyle R$ the open connected sets are $\displaystyle (-\infty ,a),~(b, \infty ),\text{ or }(a,b)$.
If two connected set have a point in common there union is connected and has one of those three forms.
Now by definition of $\displaystyle I_x$ is a maximum connected open set.
Thus, either $\displaystyle I_x=I_y\text{ or }I_x\cap I_y=\emptyset$.