MHB Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof)

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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
 

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The \rho_n are defined to be rational. The set of all rational numbers is countable so this subset is countable.
 
HallsofIvy said:
The \rho_n are defined to be rational. The set of all rational numbers is countable so this subset is countable.
Thanks HallsofIvy ...

Yes, the $$\rho_n$$ or $$\rho_k$$ are constructed or defined to be rational ... but the $$\rho_x$$ are also defined/constructed to be rational ... so my comments under question 1, I think, are correct ... do you agree ... am I interpreting the situation correctly ...

Can you comment on Question 2 ...?

Peter
 
No, the set of all rational numbers is countable. Any set of rational numbers is at most countable. If a set of rational numbers, $\{r_x\}$, is indexed by "x" from an uncountable set that just means that there are a lot of duplicates- there are uncountably many different "x" such that "$r_x$" are the same.
 
HallsofIvy said:
No, the set of all rational numbers is countable. Any set of rational numbers is at most countable. If a set of rational numbers, $\{r_x\}$, is indexed by "x" from an uncountable set that just means that there are a lot of duplicates- there are uncountably many different "x" such that "$r_x$" are the same.
Thanks HallsofIvy ...

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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