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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:

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 ... ?

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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