Don't understand the proof of a simple theorem

**gyro73** · October 15th 2009, 11:28 AM

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?

**Plato** · October 15th 2009, 12:55 PM

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

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