Ext(A) and Its Relation to Int(X-A) and Cl(A): A Definition

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In summary, the conversation discusses the statement that Ext(ExtA) = intA, and whether or not it is true. The participants consider different examples and try to prove the statement, but ultimately conclude that it is false. They also mention the importance of not blindly trusting authority and using critical thinking.
  • #1
mathboy
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Definition: Ext(A) = int (X-A) = X - Cl(A)

Is it true that Ext(ExtA) = intA ?
 
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  • #2
Yes. Can you prove it?
 
  • #3
So far, I've only proven that Ext(ExtA) c X-extA. I can't arrive at Ext(ExtA) c intA.
Are you sure the statement is true?
 
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  • #4
If x is contained in Int(A) then there is an open neighborhood of x that is not in Ext(A). Can you show that makes it a member of Ext(Ext(A))?
 
  • #5
X=R, A = rationals.

Then ext(extA)=ext(int(irrationals))= ext (empty) = R

But int(A) = empty. The statement is false.
 
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  • #6
Ext(rationals)=empty. Ext(empty)=R. Int(rationals)=empty. Yep. Good point. Not trusting authority is a good thing. Keep it up. Sorry, not thinking.
 
  • #7
I thought it was true too when I pictured A being a circle in the xy-plane. I was wondering why it was so hard to prove it.
 

FAQ: Ext(A) and Its Relation to Int(X-A) and Cl(A): A Definition

1. What is Ext(A)?

Ext(A) refers to the exterior of set A, also known as the complement of A. It includes all the elements that are not in A.

2. How is Ext(A) related to Int(X-A)?

Ext(A) and Int(X-A) are complementary sets, meaning that they contain opposite elements. Ext(A) includes all the elements that are not in A, while Int(X-A) includes all the elements that are in X but not in A.

3. What is the definition of Cl(A)?

Cl(A) refers to the closure of set A, which includes all the elements in A as well as its limit points. In other words, Cl(A) is the smallest closed set that contains A.

4. How is Ext(A) related to Cl(A)?

Ext(A) and Cl(A) are complementary sets. Ext(A) includes all the elements that are not in A, while Cl(A) includes all the elements in A and its limit points.

5. Why is understanding Ext(A) and its relation to Int(X-A) and Cl(A) important?

Understanding these concepts is crucial in topology and set theory, as they allow us to define and analyze sets and their properties. It also helps us understand the relationship between different sets and how they interact with each other.

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