Ext(ExtA) = intA ?

1. Mar 10, 2008

mathboy

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

Is it true that Ext(ExtA) = intA ?

2. Mar 10, 2008

Dick

Yes. Can you prove it?

3. Mar 10, 2008

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?

Last edited: Mar 11, 2008
4. Mar 11, 2008

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))?

5. Mar 11, 2008

mathboy

X=R, A = rationals.

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

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

Last edited: Mar 11, 2008
6. Mar 11, 2008

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.

7. Mar 11, 2008

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.