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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
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