Boundaries in topological space

[Xerxes]

Say I have a set X and a topology T on X = {X {} A} i.e A is an open subset of T. Then the complement of A is X - A = Ac, which is closed.

 

Now the interior of A, int(A) is the largest open set (or the union of all open sets) contained in A which is A, and the closure of A, cl(A) is the smallest closed set in {X {} Ac} containing A which is X. So if bd(A) = cl(A) - int(A), we have that bd(A) = X - A = Ac.

 

Similarly, the closure of Ac is the smallest closed set containing Ac, which is Ac = X - A. So, by an alternative definition of the boundary of A,

cl(A) intersect cl(Ac) = X intersect (X - A) which is X - A = Ac.

 

I've tried it out on a number of arbitrary topologies of my own devising, and the answer is always the same, except where sets in T are both open and closed or neither. Surely it's not right in general, though?

 

Hmm. Does that make sense? (just back from the pub)

[Dave]

Say I have a set X and a topology T on X = {X {} A} i.e A is an open subset of T. Then the complement of A is X - A = Ac, which is closed.

 

I'm assuming that this means [imath]\mathcal{T} = \left\{ X, \phi, A \right\}[/imath] for some [imath]A \subset X[/imath]? Or am I being silly?

[Xerxes]

I'm assuming that this means [imath]\mathcal{T} = \left\{ X, \phi, A \right\}[/imath] for some [imath]A \subset X[/imath']? Or am I being silly?

Yes that's exactly what I meant! Was there any ambiguity in not bothering with LaTeX? As to substance......?

[Dave]

I can't see an error in your reasoning. I'll have a look at it tomorrow when I'm not so tired