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

[mathboy]

Definition: Ext(A) = int (X-A) = X - Cl(A)

Is it true that Ext(ExtA) = intA ?

 

[Dick]

Yes. Can you prove it?

 

[mathboy]

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?

 

[Dick]

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))?

 

[mathboy]

X=R, A = rationals.

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

But int(A) = empty. The statement is false.

 

[Dick]

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.

 

[mathboy]

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.