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- Jun 22, 2012

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I need help in order to fully understand Example 2.7(a) ... ..

The relevant text reads as follows:

My questions are as follows:

**Question 1**In the above example from Willard we read the following;

" ... ... If \(\displaystyle A\) is an open set in \(\displaystyle \mathbb{R}\), the relation \(\displaystyle x \sim y\) iff there is some open interval \(\displaystyle (a, b)\) with \(\displaystyle \{ x, y \} \subset (a, b) \subset A\) is an equivalence relation on \(\displaystyle A\) and the resulting equivalence classes are disjoint open intervals whose union is \(\displaystyle A\) ... ... "

Can someone please demonstrate formally and rigorously that the resulting equivalence classes are disjoint open intervals whose union is \(\displaystyle A\) ... ... ?

**Question 2**

In the above example from Willard we read the following;

" ... ... The fact that there can be only countably many follows since each must contain a distinct rational ... "

I am somewhat lost in trying to understand this statement ... can someone please explain the meaning of "there can be only countably many follows since each must contain a distinct rational ...?

Help will be much appreciated ... ...

Peter