Re: Closing the Intersection

On Tue, 25 Apr 2006, William Elliot wrote:

From: Jose Capco <cliomseerg@xxxxxxxxxxxxxxxxxxxxxxxxx>
William Elliot wrote:

Huh? Some x in cl A/\B with x not in cl A and x not in cl B ?
How is that possible?
x in cl A/\B subset cl A /\ cl B subset cl A, cl B.

Sorry I meant x in cl(A) /\ cl(B) but not in cl(A/\B),
whilst A and B being connected with A/\B nonempty.

So what entirely, is your question?

Frequently I download replies instead of reading online. Thus
in attempts to compensate for your over clipping I'll surmise
you're speculating if A,B are connected, that
cl A/\B = cl A /\ cl B.

Did not my counterexample deter you from that conjecture?

Now a real plane counter example.
Let A be vertical lines with rational x coordinate and B
same with irrational x coordinate. Add x axis to both.
cl A/\B = x axis; cl A /\ cl B = R^2;

Very plane counter example with locally connected
and simply connected open sets and common intersection.

A = plane with closed first quadrant removed
B = plane with closed second quadrant removed

cl(A /\ B) = closed 3rd and 4th quadrants
cl A /\ cl B = cl(A /\ B) \/ y-axis

.