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

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The discussion centers on the relationship between Ext(A), Int(X-A), and Cl(A), asserting that Ext(A) equals Int(X-A) and X minus Cl(A). A participant questions the validity of the statement Ext(ExtA) equating to Int(A) and seeks proof, noting that they have only shown Ext(ExtA) is a subset of X minus ExtA. Another contributor points out that if x is in Int(A), it leads to a contradiction regarding membership in Ext(Ext(A)). The conversation reveals confusion around the implications of these definitions, particularly when considering specific sets like the rationals and irrationals in R. Ultimately, the group acknowledges the complexity of the concepts and the importance of questioning assumptions in mathematical proofs.
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Definition: Ext(A) = int (X-A) = X - Cl(A)

Is it true that Ext(ExtA) = intA ?
 
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Yes. Can you prove it?
 
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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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))?
 
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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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.
 
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.
 

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