Re: Closing the Intersection

On Fri, 21 Apr 2006, Jose Capco wrote:
William Elliot wrote:
On Fri, 21 Apr 2006, Jose Capco wrote:

Let X be a topological space. Is there a characterization that tells us
when a given A,B subset of X can have

cl(A /\ B)=cl(A)/\cl(B)
When A and B are closed.

I don't believe that is a characterization. Its only a necessary
condition. There are A and Bs in some topologies by which the equality
holds without either of them beling closed.

It's not a necessary and sufficient condition.
For example when A subset B

Are you wanting a property p for which for all A,B
p(A) & p(B) iff cl(A /\ B) = cl A /\ cl B ?
That is not possible. p(nulset) and for all A,
cl(nulset /\ A) = cl nulset /\ cl A
Thus for all A, p(A), which cannot be.

Hence you want property p for which for all A,B
p(A,B) iff cl A/\B = cl A /\ cl B ?
Then let p(A,B) be cl A/\B = cl A /\ cl B.
or perferable take p(A,B) as
cl A /\ cl B subset cl A/\B
Otherwise consider cases
A subset B
A,B disjoint
other
For example,
A subset B or B subset A ==> p(A,B)
A,B closed ==> p(A,B)
p(A,S\A) ==> A clopen
A,B completely separated, ie disjoint closures ==> p(A,B)
Hm, p(A,B) iff p(B,A); p(A,A).

Perhaps you'd content yourself with property p weaker than closed
for which
p(A), p(B) ==> cl A/\B = cl A /\ cl B.

Have you a clarification of characterization?
Does
cl A /\ cl B subset cl A/\B
suffice?
.

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