Open Subsets in a Metric Space .... Stromberg, Theorem 3.6 ... ....

In summary: Updated - - -In summary, the conversation involves a request for help understanding Theorem 3.6 on page 94 of Karl R. Stromberg's book, "An Introduction to Classical Real Analysis". The conversation also touches upon the definitions of open sets and open balls, and how the proof of Theorem 3.6 relies on these concepts. The summary also includes a demonstration of why letting r = min{r1, r2, ..., rn} implies that B_r(a) is a subset of U_j for each j = 1,2,...,n.
  • #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 help in order to fully understand the proof of Theorem 3.6 on page 94 ... ... Theorem 3.6 and its proof read as follows:
View attachment 9114
In the above proof by Stromberg we read the following:

" ... ... Letting \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) we see that \(\displaystyle B_r (a) \subset U_j \text{ for each } j = 1,2, \ ... \ ... n\) ... ... "Although it seems plausible ... I do not see ... rigorously speaking, why the above statement is true ...

Can someone demonstrate rigorously that letting \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) ...

... implies that \(\displaystyle B_r (a) \subset U_j\) for each \(\displaystyle j = 1,2, \ ... \ ... n\) ... ... Surely it is possible that \(\displaystyle B_r (a)\) lies partly outside some \(\displaystyle U_j\) ... ...

Help will be appreciated ...

Peter
 

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  • #2
The Theorem makes reference to "Definition 3.3" but you don't give us that definition. Also you are asking about "[tex]B_r(a)[/tex]" but don't tell us what that means. Since the theorem is about "open sets", I strongly suspect, but can't be sure, that "Definition 3.3" is the definition of "open set" and that "[tex]B_r(a)[/tex]" is the "open ball" centered at a with radius r.

If that is correct then [tex]B_r(a)[/tex] is [tex]\{ p| d(p, a)< r\}[/tex], the set of all points whose distance from point a (the center of the ball) is less than r (the radius of the ball). From that it follows immediately that if [tex]r_1< r_2[/tex] then [tex]B_{r_1}(a)\subset B_{r_2}(a)[/tex]- the smaller radius ball is inside the larger radius ball.
 
  • #3
Peter said:
Can someone demonstrate rigorously that letting \(\displaystyle r = \text{min} \{ r_1, r_2, \ ... \ ... \ r_n \}\) ...

... implies that \(\displaystyle B_r (a) \subset U_j\) for each \(\displaystyle j = 1,2, \ ... \ ... n\) ... ...
$r$ is the minimum of all the $r_j$ and so $r\leqslant r_j$ for all $j$; hence $B_r(a)\subseteq B_{r_j}(a)\subset U_j$ for all $j=1,\ldots,n$.
 
  • #4
Olinguito said:
$r$ is the minimum of all the $r_j$ and so $r\leqslant r_j$ for all $j$; hence $B_r(a)\subseteq B_{r_j}(a)\subset U_j$ for all $j=1,\ldots,n$.
Thanks for the help Olinguito ...

Peter

- - - Updated - - -

HallsofIvy said:
The Theorem makes reference to "Definition 3.3" but you don't give us that definition. Also you are asking about "[tex]B_r(a)[/tex]" but don't tell us what that means. Since the theorem is about "open sets", I strongly suspect, but can't be sure, that "Definition 3.3" is the definition of "open set" and that "[tex]B_r(a)[/tex]" is the "open ball" centered at a with radius r.

If that is correct then [tex]B_r(a)[/tex] is [tex]\{ p| d(p, a)< r\}[/tex], the set of all points whose distance from point a (the center of the ball) is less than r (the radius of the ball). From that it follows immediately that if [tex]r_1< r_2[/tex] then [tex]B_{r_1}(a)\subset B_{r_2}(a)[/tex]- the smaller radius ball is inside the larger radius ball.

Thanks for the help HallsofIvy ...

Peter
 

Related to Open Subsets in a Metric Space .... Stromberg, Theorem 3.6 ... ....

1. What is an open subset in a metric space?

An open subset in a metric space is a subset of the metric space in which every point has a neighborhood contained entirely within the subset. In other words, for every point in the open subset, there exists a small enough radius such that all points within that radius are also in the subset.

2. How is an open subset different from a closed subset?

An open subset is different from a closed subset in that a closed subset contains all of its limit points, while an open subset does not necessarily contain all of its limit points. Additionally, a closed subset may have a boundary, while an open subset does not have a boundary.

3. What is the significance of Stromberg, Theorem 3.6 in relation to open subsets in a metric space?

Stromberg, Theorem 3.6 states that in a metric space, every open subset can be written as a union of a countable collection of open balls. This theorem is significant because it provides a way to represent open subsets in a metric space in a more manageable and countable way.

4. Can an open subset have an infinite number of points?

Yes, an open subset can have an infinite number of points. This is because an open subset can contain all of its limit points, which can be an infinite number of points. However, it is also possible for an open subset to have a finite number of points.

5. How are open subsets in a metric space used in mathematical analysis?

Open subsets in a metric space are used in mathematical analysis to define continuity, differentiability, and other important concepts. They are also used in the definition of open and closed sets, which are fundamental in topology and measure theory. Additionally, open subsets are used in the study of convergence and limits of sequences and series in metric spaces.

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