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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:Question 1
In the above example from Willard we read the following;
" ... ... If $$A$$ is an open set in $$\mathbb{R}$$, the relation $$x \sim y$$ iff there is some open interval $$(a, b)$$ with $$\{ x, y \} \subset (a, b) \subset A$$ is an equivalence relation on $$A$$ and the resulting equivalence classes are disjoint open intervals whose union is $$A$$ ... ... "Can someone please demonstrate formally and rigorously that the resulting equivalence classes are disjoint open intervals whose union is $$A$$ ... ... ?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
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 $$A$$ is an open set in $$\mathbb{R}$$, the relation $$x \sim y$$ iff there is some open interval $$(a, b)$$ with $$\{ x, y \} \subset (a, b) \subset A$$ is an equivalence relation on $$A$$ and the resulting equivalence classes are disjoint open intervals whose union is $$A$$ ... ... "Can someone please demonstrate formally and rigorously that the resulting equivalence classes are disjoint open intervals whose union is $$A$$ ... ... ?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