Lindelof Covering Theorem .... Apostol, Theorem 3.28 ....

[Math Amateur]

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...

I am focused on Chapter 3: Elements of Point Set Topology ... ...

I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows:View attachment 9082
View attachment 9083In the above proof by Apostol we read the following:

" ... ... The set of all \(\displaystyle n\)-balls \(\displaystyle A_{ m(x) }\) obtained as \(\displaystyle x\) varies over all elements of \(\displaystyle A\) is a countable collection of open sets which covers \(\displaystyle A\) ... ..."
My question is as follows:

What happens when \(\displaystyle A\) is an uncountably infinite set ... how does the set of all \(\displaystyle n\)-balls \(\displaystyle A_{ m(x) }\) remain as a countable collection of open sets which covers \(\displaystyle A\) ... when \(\displaystyle x\) ranges over an uncountable set ... ...?My thoughts are as follows: ... ... ... the sets \(\displaystyle A_{ m(x) }\) must be used many times ... indeed in many cases infinitely many times ... is that correct?

Help will be much appreciated ...

Peter=====================================================================================The above post refers to Theorem 3.27 ... so I am providing text of the same ... as follows:View attachment 9084
View attachment 9085
Hope that helps ...

Peter

 

[Opalg]

In the above proof by Apostol we read the following:

" ... ... The set of all \(\displaystyle n\)-balls \(\displaystyle A_{ m(x) }\) obtained as \(\displaystyle x\) varies over all elements of \(\displaystyle A\) is a countable collection of open sets which covers \(\displaystyle A\) ... ..."

My question is as follows:

What happens when \(\displaystyle A\) is an uncountably infinite set ... how does the set of all \(\displaystyle n\)-balls \(\displaystyle A_{ m(x) }\) remain as a countable collection of open sets which covers \(\displaystyle A\) ... when \(\displaystyle x\) ranges over an uncountable set ... ...?

My thoughts are as follows: ... ... ... the sets \(\displaystyle A_{ m(x) }\) must be used many times ... indeed in many cases infinitely many times ... is that correct?

That is correct. There are uncountably many points in $A$, but there are only countably many elements $A_k$ in $G$. So (in general) there will be uncountably many different points $x\in A$ giving rise to the same element $A_k = A_{m(x)}\in G$.

I very much prefer Apostol's proof of the Lindelöf covering theorem to that of Sohrab which you quoted in https://mathhelpboards.com/analysis-50/compact-subsets-r-sohrab-proposition-4-1-1-lindelof-26249.html#post115993. I found Sohrab's proof almost impenetrable, but Apostol presents essentially the same argument in a much more transparent way.

 

[Math Amateur]

That is correct. There are uncountably many points in $A$, but there are only countably many elements $A_k$ in $G$. So (in general) there will be uncountably many different points $x\in A$ giving rise to the same element $A_k = A_{m(x)}\in G$.

I very much prefer Apostol's proof of the Lindelöf covering theorem to that of Sohrab which you quoted in https://mathhelpboards.com/analysis-50/compact-subsets-r-sohrab-proposition-4-1-1-lindelof-26249.html#post115993. I found Sohrab's proof almost impenetrable, but Apostol presents essentially the same argument in a much more transparent way.

Thanks for the help Opalg ...

The fact that you found Sohrab's proof almost impenetrable was such a relief to me ... since I found it. Incredibly difficult/impossible ... but I did follow Apostol ...

Thanks again ...

Peter