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

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Homework Help Overview

The discussion revolves around the definitions and relationships of the concepts Ext(A), Int(X-A), and Cl(A) within the context of set theory and topology, particularly focusing on the properties of sets in the real numbers.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the validity of the statement Ext(ExtA) = Int(A) and discuss various proofs and counterexamples. Questions arise regarding the implications of certain definitions and the relationships between the sets involved.

Discussion Status

The discussion is ongoing, with some participants providing insights and examples that challenge the original statement. There is a recognition of differing interpretations and attempts to clarify the relationships between the sets, but no consensus has been reached.

Contextual Notes

Participants reference specific examples, such as the rationals and irrationals in the real numbers, to illustrate their points. There is an acknowledgment of the complexity of the definitions and the need for careful consideration of the assumptions involved.

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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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?
 
Last edited:
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.
 
Last edited:
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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