# Equivalent Statements to Compactness ... Another Question ... Stromberg, Theorem 3.43 ... ...

#### Peter

##### Well-known member
MHB Site Helper
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:

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

Last edited:

#### Opalg

##### MHB Oldtimer
Staff member
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.