# Don't understand the proof of a simple theorem

• Oct 15th 2009, 11:28 AM
gyro73
Don't understand the proof of a simple theorem
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?
• Oct 15th 2009, 12:55 PM
Plato
Quote:

Originally Posted by gyro73
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.

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 $\displaystyle I_x=I_y\text{ or }I_x\cap I_y=\emptyset$.