Understanding Theorem 3.43: Comparing U_B & V_k in Stromberg's Book

[Math Amateur]

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need further help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ... Theorem 3.43 and its proof read as follows:

View attachment 9149
View attachment 9150

At about the middle of the above proof by Stromberg we read the following:

" ... ... Otherwise enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ V_k \}_{ k = 1 }^{ \infty }\). ... ... "I am wondering what are the \(\displaystyle V_k\) ... are they elements of \(\displaystyle \mathscr{V}\) (... that is, the \(\displaystyle U_B\)) ... or are they sets of some kind ... ... can someone please explain and elucidate the nature of the \(\displaystyle V_k\) ...

... indeed maybe the \(\displaystyle V_k\) are just equal to the \(\displaystyle U_B\) ... in that case why not enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) ...

Hope someone can help ...

Peter

 

[Opalg]

At about the middle of the above proof by Stromberg we read the following:

" ... ... Otherwise enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ V_k \}_{ k = 1 }^{ \infty }\). ... ... "I am wondering what are the \(\displaystyle V_k\) ... are they elements of \(\displaystyle \mathscr{V}\) (... that is, the \(\displaystyle U_B\)) ... or are they sets of some kind ... ... can someone please explain and elucidate the nature of the \(\displaystyle V_k\) ...

... indeed maybe the \(\displaystyle V_k\) are just equal to the \(\displaystyle U_B\) ... in that case why not enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) ...

I haven't read this proof carefully, but I am sure that you are correct: the elements of \(\displaystyle \mathscr{V}\) are exactly the sets \(\displaystyle U_B\), and it would be quite permissible to enumerate them as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) (though that implies that you have enumerated the sets \(\displaystyle U_B\)). I think that Stromberg found it better to enumerate the sets in \(\displaystyle \mathscr{V}\) directly, rather than indirectly by enumerating the sets in $\mathscr{B}$. In that way, he avoids cumbersome double subscripts.

 

[soloenergy]

First of all, it's great that you are reading Stromberg's book and focusing on Chapter 3. Limits and continuity can be a challenging topic, so it's understandable that you may need some extra help understanding Theorem 3.43.

To answer your question, V_k are indeed elements of \mathscr{V}, which is the collection of all open sets in the space X. In other words, they are sets of some kind. The reason for enumerating them as V_k instead of U_{B_k} is simply a matter of notation and preference. Both notations are commonly used in mathematics, so it's up to the author's discretion which one they choose to use in their proof.

In this case, Stromberg has chosen to use V_k to represent the elements of \mathscr{V}, which is a common notation for indexed sets. It's important to note that the notation V_k does not necessarily mean that the sets are equal to U_B. V_k is just a way to refer to the elements of \mathscr{V} in the proof.

I hope this helps clarify the nature of V_k in the proof of Theorem 3.43. If you have any further questions or need more clarification, don't hesitate to ask. Good luck with your studies!