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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 $$\mathbb{R}$$ 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 $$\{ \rho_x \ : \ x \in O \} \subset \mathbb{Q}$$ is countable ... ... "
But ... it seems to me that since the $$x$$'s are uncountable that the number of $$\rho_x$$ 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 $$B_{ \rho_x } (x)$$ ... 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 $$k \in \mathbb{N}$$ we pick $$\lambda_k \in B_{ \rho_k } (x_k) \subset O_{ \lambda_k }$$, then we have a countable subcollection $$\{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }$$ which satisfies $$O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }$$ ... ..."Can someone please explain/demonstrate clearly (preferably in some detail) how the process described actually results in a countable subcollection where $$O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }$$ ...
... indeed, given that $$\{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }$$
... it looks as if $$\bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k } \subset O$$ ... ?
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 $$\mathbb{R}$$ 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 $$\{ \rho_x \ : \ x \in O \} \subset \mathbb{Q}$$ is countable ... ... "
But ... it seems to me that since the $$x$$'s are uncountable that the number of $$\rho_x$$ 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 $$B_{ \rho_x } (x)$$ ... 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 $$k \in \mathbb{N}$$ we pick $$\lambda_k \in B_{ \rho_k } (x_k) \subset O_{ \lambda_k }$$, then we have a countable subcollection $$\{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }$$ which satisfies $$O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }$$ ... ..."Can someone please explain/demonstrate clearly (preferably in some detail) how the process described actually results in a countable subcollection where $$O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }$$ ...
... indeed, given that $$\{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }$$
... it looks as if $$\bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k } \subset O$$ ... ?
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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