Why is \( c'' \) an Upper Bound for \( U \cap [a, b] \) in Theorem 3.47?

In summary, Peter is reading Karl R. Stromberg's book "An Introduction to Classical Real Analysis" and is focused on Chapter 3: Limits and Continuity. He needs further help understanding the proof of Theorem 3.47 on page 107, specifically the third paragraph. He asks for a rigorous demonstration of why $U \cap V \cap [a, b] \subset U \cap V \cap S = \emptyset \Longrightarrow c''$ is an upper bound for $U \cap [a, b]$. GJA provides a helpful response, explaining that by the definition of $c$, $[p,c] \cap (U \cap [a, b]) \neq \emptyset$
  • #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 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:

View attachment 9155In the third paragraph of the above proof by Stromberg we read the following:

" ... ... But \(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset\), and so \(\displaystyle c''\) is an upper bound for \(\displaystyle U \cap [a, b]\) ... ... " My question is as follows:

Can someone please demonstrate rigorously how/why ...

\(\displaystyle U \cap V \cap [a, b] \ \subset \ U \cap V \cap S = \emptyset \Longrightarrow c''\) is an upper bound for \(\displaystyle U \cap [a, b]\) ...
Help will be appreciated ...

Peter
 

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  • #2
Hi Peter,

By the definition of $c$, $[p,c]\cap\left(U\cap [a,b]\right)\neq\emptyset$ for all $p<c.$ However, $c''<c$ and, by the choice for $c''$, $$[c'',c]\cap\left(U\cap [a,b]\right)\subset U\cap V\cap [a,b]=\emptyset.$$ Hence, $[c'',c]\cap\left(U\cap [a,b]\right)=\emptyset,$ contradicting the definition of $c$.
 
  • #3
GJA said:
Hi Peter,

By the definition of $c$, $[p,c]\cap\left(U\cap [a,b]\right)\neq\emptyset$ for all $p<c.$ However, $c''<c$ and, by the choice for $c''$, $$[c'',c]\cap\left(U\cap [a,b]\right)\subset U\cap V\cap [a,b]=\emptyset.$$ Hence, $[c'',c]\cap\left(U\cap [a,b]\right)=\emptyset,$ contradicting the definition of $c$.
Thanks for your reply, GJA ...

It was most helpful ...

Peter
 

Related to Why is \( c'' \) an Upper Bound for \( U \cap [a, b] \) in Theorem 3.47?

1. What is connectedness in R?

Connectedness in R refers to the property of a topological space where there are no disjoint open sets that cover the space. In simpler terms, a space is connected if it cannot be broken into two or more separate pieces.

2. How is connectedness determined in R?

In R, connectedness can be determined by examining the open sets in the space and determining if they are disjoint or not. If there are no disjoint open sets, then the space is connected.

3. What are intervals in R?

In R, intervals refer to a range of values between two points on a number line. They can be open (not including the endpoints), closed (including the endpoints), or half-open (including one endpoint but not the other).

4. How are intervals represented in R?

In R, intervals can be represented using the notation [a, b] for closed intervals, (a, b) for open intervals, and [a, b) or (a, b] for half-open intervals.

5. What is Stromberg, Theorem 3.47 in R?

Stromberg, Theorem 3.47 is a theorem in R that states that if a set in R is both open and closed, then the set must be either the entire space or the empty set. This is known as the connectedness theorem.

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