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?

Printable View

- Oct 15th 2009, 11:28 AMgyro73Don'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 PMPlato
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 $\displaystyle I_x=I_y\text{ or }I_x\cap I_y=\emptyset$.