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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 4: Topology of \(\displaystyle \mathbb{R}\) and Continuity ... ...
I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read as follows:
View attachment 9081
My questions are as follows:Question 1
In the above proof by Sohrab we read the following:
" ... ... Now the set \(\displaystyle \{ \rho_x \ : \ x \in O \} \subset \mathbb{Q}\) is countable ... ... "
But ... it seems to me that since the \(\displaystyle x\)'s are uncountable that the number of \(\displaystyle \rho_x\) is uncountable ... but that many (at times infinitely many ... ) have the same values since each is equal to a rational number and these are countable ...
... so in fact there are an uncountably infinite number of open balls \(\displaystyle B_{ \rho_x } (x)\) ... there are just a countable number of different values for the radii of the open balls ...
Is my interpretation correct ... ?
Question 2
In the above proof by Sohrab we read the following:
" ... ... If for each \(\displaystyle k \in \mathbb{N}\) we pick \(\displaystyle \lambda_k \in B_{ \rho_k } (x_k) \subset O_{ \lambda_k }\), then we have a countable subcollection \(\displaystyle \{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }\) which satisfies \(\displaystyle O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }\) ... ..."Can someone please explain/demonstrate clearly (preferably in some detail) how the process described actually results in a countable subcollection where \(\displaystyle O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }\) ...
... indeed, given that \(\displaystyle \{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }\)
... it looks as if \(\displaystyle \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k } \subset O\) ... ?
In addition to answers to the two questions, any explanations/clarifications of the overall strategy and tactics of the proof would be very gratefully received ...
Help will be much appreciated ...
Peter
I am focused on Chapter 4: Topology of \(\displaystyle \mathbb{R}\) and Continuity ... ...
I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read as follows:
View attachment 9081
My questions are as follows:Question 1
In the above proof by Sohrab we read the following:
" ... ... Now the set \(\displaystyle \{ \rho_x \ : \ x \in O \} \subset \mathbb{Q}\) is countable ... ... "
But ... it seems to me that since the \(\displaystyle x\)'s are uncountable that the number of \(\displaystyle \rho_x\) is uncountable ... but that many (at times infinitely many ... ) have the same values since each is equal to a rational number and these are countable ...
... so in fact there are an uncountably infinite number of open balls \(\displaystyle B_{ \rho_x } (x)\) ... there are just a countable number of different values for the radii of the open balls ...
Is my interpretation correct ... ?
Question 2
In the above proof by Sohrab we read the following:
" ... ... If for each \(\displaystyle k \in \mathbb{N}\) we pick \(\displaystyle \lambda_k \in B_{ \rho_k } (x_k) \subset O_{ \lambda_k }\), then we have a countable subcollection \(\displaystyle \{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }\) which satisfies \(\displaystyle O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }\) ... ..."Can someone please explain/demonstrate clearly (preferably in some detail) how the process described actually results in a countable subcollection where \(\displaystyle O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }\) ...
... indeed, given that \(\displaystyle \{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }\)
... it looks as if \(\displaystyle \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k } \subset O\) ... ?
In addition to answers to the two questions, any explanations/clarifications of the overall strategy and tactics of the proof would be very gratefully received ...
Help will be much appreciated ...
Peter
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