# Closure of the Rational Numbers (Using Standard Topology)

• Fluffman4
In summary, to prove that Cl(Q) = R in the standard topology, you need to show that the closure of the set of rational numbers, denoted as Cl(Q), is equal to the set of real numbers, R. This can be done by defining the closure of a subset A of a topological space as the intersection of all closed sets containing A, and finding the smallest closed set containing A for closure. Additionally, it is important to note that any interval of real numbers contains both rational and irrational numbers.
Fluffman4
Prove that Cl(Q) = R in the standard topology
I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.

You should state the definition of 'Cl(Q)' you are using. Is it the intersection of all closed sets containing Q? If so, think about what the complement of a closed set containing Q must be.

We defined the closure of a subset A of a topological space as the intersection of all closed sets containing A. We basically want to find the smallest closed set containing A for closure.

Fluffman4 said:
We defined the closure of a subset A of a topological space as the intersection of all closed sets containing A. We basically want to find the smallest closed set containing A for closure.

Ok. So describe a closed set A in R containing Q. You might find it easier to describe the complement of A.

Crucial point: given any interval of real numbers, there exist both rational and irrational numbers in that interval.

## 1. What is the closure of the rational numbers?

The closure of the rational numbers refers to the set of all real numbers that can be reached by taking limit points of the rational numbers. This includes the rational numbers themselves, as well as any irrational numbers that can be approached by a sequence of rational numbers.

## 2. How is the closure of the rational numbers defined using standard topology?

In standard topology, the closure of a set is defined as the union of the set and all of its limit points. So, in the case of the rational numbers, the closure would be the set of all rational numbers and any real numbers that can be approached by a sequence of rational numbers.

## 3. Why is the closure of the rational numbers important?

The closure of the rational numbers is important because it helps to fill in the gaps between rational numbers and allows us to have a complete understanding of the real numbers. It also helps in the study of continuity and convergence in analysis.

## 4. Is the closure of the rational numbers the same as the closure of the integers?

No, the closure of the rational numbers is not the same as the closure of the integers. While the closure of the rational numbers includes both the rational and irrational numbers, the closure of the integers only includes the integers themselves.

## 5. How is the closure of the rational numbers related to the Cantor set?

The Cantor set is a perfect example of a subset of the real numbers that is uncountable and has a measure of zero. It can be shown that the Cantor set is contained within the closure of the rational numbers, as it includes all points that can be approached by a sequence of rational numbers. Therefore, the closure of the rational numbers is a larger set that contains the Cantor set.

• Calculus and Beyond Homework Help
Replies
2
Views
496
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
24
Views
2K
• Calculus and Beyond Homework Help
Replies
10
Views
1K
• Linear and Abstract Algebra
Replies
11
Views
2K
• Calculus and Beyond Homework Help
Replies
15
Views
15K
• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
2K