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

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

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

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  • Apostol - 1- Theorem 3.27 ... PART 1 ... .png
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Peter said:
In the above proof by Apostol we read the following:

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

My question is as follows:

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

My thoughts are as follows: ... ... ... the sets $$A_{ m(x) }$$ 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.
 
Opalg said:
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
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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