Exercise 2.32.b

Exercise 2.32.b: Give $$\omega=\{0,1,2,\dots\}$$ the cofinite topology. Let $$A=\{0,10,100,1000,1001,1002,1003,\dots\}$$. Find $$A',Cl(A),Int(A),Bd(A)$$.

Tony Bruguier
The set A is open, so int(A)=A. (Yep! --Steven.clontz 15:59, March 24, 2011 (UTC))

To know A' I just have to look at whether 0 or 1 or both are limit points. All the other cases reduce to 0 or 1. Being in the topology means "you take w and remove a finite number of points." But the intersection of A with any of those sets always has at least two elements in them (in fact infinity, I think), so both 0 and 1 are limits and A'=w. Thus Cl(A)=w. (You're right, but you need a slightly more precise argument. Say, for each $$n\in\omega$$, show that an open set $$U$$ containing $$n$$ must intersect $$A$$ somewhere. --Steven.clontz 15:59, March 24, 2011 (UTC))

For the boundary, I am not sure about definition 2.30. It doesn't say whether the point in A (resp. Ac) must be different from p. The definition doesn't say it must be different, so I will assume that they can be the same. (Yes, in general you should always do this. --Steven.clontz 15:59, March 24, 2011 (UTC)) Once again, I can look at 0 and 1 only. The point 0 is not in the border because you can choose A as the open set. By similar reasoning as paragraph above, the intersection of two open sets has at least one point and so 1 is a border point. Thus Bd(A)=Ac. [If we impose different, the Bd(A)=empty set). '''(You have the correct answer, but again, to be precise, you should show that if $$n \in A$$, there is an open set which contains no point of $$A^c$$, and if you have $$n \in A^c$$, every open set also intersects $$A$$ somewhere. --Steven.clontz 15:59, March 24, 2011 (UTC))'''

Please review this. I am really not sure. If this is correct, I can put in mathy language. Tony Bruguier 02:43, March 24, 2011 (UTC)