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

In summary, V_k represents the elements of \mathscr{V}, which is the collection of open sets in the space X. It is a matter of notation and preference to use V_k instead of U_{B_k} in the proof of Theorem 3.43.
  • #1
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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:

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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
 

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  • #2
Peter said:
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.
 
  • #3

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!
 

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

1. What is Theorem 3.43 in Stromberg's Book?

Theorem 3.43 in Stromberg's Book is a mathematical theorem that compares two different mathematical objects, U_B and V_k, in terms of their properties and relationships.

2. What is the significance of Theorem 3.43 in Stromberg's Book?

Theorem 3.43 is significant because it provides a deeper understanding of the properties and relationships between U_B and V_k, which are important mathematical objects in the field of mathematics.

3. How is Theorem 3.43 useful in mathematics?

Theorem 3.43 is useful in mathematics because it helps mathematicians compare and analyze the properties and relationships of U_B and V_k, which can lead to new discoveries and insights in various mathematical fields.

4. Can Theorem 3.43 be applied in real-world situations?

Yes, Theorem 3.43 can be applied in real-world situations where U_B and V_k are relevant mathematical objects. For example, it can be used in physics to compare different physical quantities or in economics to compare different economic models.

5. Are there any prerequisites for understanding Theorem 3.43?

Yes, a basic understanding of mathematical concepts and notation is necessary to fully comprehend Theorem 3.43. Familiarity with the properties and relationships of U_B and V_k will also be helpful in understanding the theorem.

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