Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3 ....

In summary, the conversation discusses help with a specific exercise from the book "Basic Real Analysis" by Houshang H. Sohrab. The exercise involves proving the closedness of the sets of natural and integer numbers using the definitions of open and closed sets. The solution involves considering the complements of these sets as unions of open intervals. The conversation also discusses the set ##\{1/n ~|~ n \in \Bbb{N} \}## and how it is neither closed nor open, using the numbers 0 and 1 as examples.
  • #1
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Homework Statement



I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with a part of Exercise 2.2.4 Part (3) ... ...

Exercise 2.2.4 Part (3) reads as follows:

?temp_hash=ae127f8cdbb2ce1c102644934fa8ac7f.png


Homework Equations



The definitions of open and closed sets are relevant as is the definition of an \epsilon neighborhood. Sohrab defines these concepts/entities as follows:

?temp_hash=ae127f8cdbb2ce1c102644934fa8ac7f.png

The Attempt at a Solution



Reflecting in general terms, I suspect the proof that ##\mathbb{N}## and ##\mathbb{Z}## are closed is approached by looking at the complements of the sets of ##\mathbb{N}## and ##\mathbb{Z}## ... visually ##\mathbb{R}## \ ##\mathbb{N}## and ##\mathbb{R}## \ ##\mathbb{Z}## and proving that these sets are open ... which intuitively they seem to be ... but I cannot see how to technically write the proof in terms of open sets and ##\epsilon##-neighbourhoods ... can someone please help ...

I have not made any progress regarding the set ##\{ \frac{1}{n} \ : \ n \in \mathbb{N} \}## ...

Peter
 

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  • #2
Pick any real number, r (use a variable, not a specific number), not in the set and determine ε such that 2ε is the distance to the nearest number in the set. Proceed from there.
 
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  • #3
FactChecker said:
Pick any real number, r (use a variable, not a specific number), not in the set and determine ε such that 2ε is the distance to the nearest number in the set. Proceed from there.
Thanks FactChecker ... I am assuming you are referring to the set ##\{ \frac{1}{n} \ : \ n \in \mathbb{N} \}## and not to the proof that ##\mathbb{N}## and ##\mathbb{Z}## are closed ... is that correct ... ?

Peter
 
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  • #4
Math Amateur said:
Thanks FactChecker ... I am assuming you are referring to the set ##\{ \frac{1}{n} \ : \ n \in \mathbb{N} \}## and not to the proof that ##\mathbb{N}## and ##\mathbb{Z}## are closed ... is that correct ... ?

Peter
For N and Z, you can use it to prove that the complement is open. For the 1/n set, you can use it to prove that the complement is not open. It is using the definitions of open and closed sets directly, which is what you want to do unless you have some other proven theorems to use.
 
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  • #5
Hi FactChecker ... I am still perplexed ...

Can you help further ...

Peter
 
  • #6
Do you know whether an arbitrary union of open sets is open; that is, if ##\{U_i\}_{i \in I}## is some collection of open intervals, do you know whether ##\bigcup_{i \in I} U_i## is open? It shouldn't be hard to prove it from the definition you provided in your OP. If so, you should be pretty straightforward to write the complement of ##\Bbb{N}## and ##\Bbb{Z}## as the union of open intervals.
 
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  • #7
As for showing the set ##\{1/n ~|~ n \in \Bbb{N} \}## is neither closed nor open, consider the numbers ##0## and ##1##. Using ##0## will help you show that it isn't closed, whereas ##1## will help you show that it isn't open.
 

Related to Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3 ....

1. What is the difference between an open set and a closed set?

An open set is a set that contains all of its interior points, meaning that every point in the set has a neighborhood that is also contained within the set. A closed set, on the other hand, contains all of its boundary points, meaning that every limit point of the set is also included in the set. In simpler terms, an open set has no boundary, while a closed set has a boundary.

2. How can you tell if a set is open or closed?

A set can be determined to be open or closed by examining the points within the set. If every point in the set has a neighborhood that is also contained within the set, then the set is open. If every limit point of the set is also included in the set, then the set is closed. It is important to note that a set can also be both open and closed, known as a clopen set.

3. What are some examples of open and closed sets?

An example of an open set would be the set of all real numbers between 0 and 1, not including 0 and 1 themselves. This set has no boundary and every point within it has a neighborhood that is also contained within the set. An example of a closed set would be the set of all integers, as it contains all of its boundary points (i.e. every limit point is also an integer).

4. Can a set be both open and closed?

Yes, a set can be both open and closed. These sets are known as clopen sets. An example of a clopen set would be the set of all rational numbers, as it contains all of its limit points (irrational numbers) while also having no boundary.

5. Why is it important to understand open and closed sets?

Understanding open and closed sets is important in many areas of mathematics, such as topology and real analysis. It allows us to define and classify different types of sets, which can be useful in solving problems and proving theorems. Additionally, the concepts of open and closed sets are often used in other fields, such as physics and computer science, to study complex systems and algorithms.

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