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I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 1: Set Theory and Metric Spaces and am currently focused on Section 2: Metric Spaces ... ...

I need help in order to fully understand Example 2.7(a) ... .. The relevant text reads as follows:View attachment 9688My questions are as follows:

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 \(\displaystyle A\) ... ... ?

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

I need help in order to fully understand Example 2.7(a) ... .. The relevant text reads as follows:View attachment 9688My 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