Basic Topology, a little set theory ?

Question

What is an example of how the "(closure of A) intersect (closure of B)" is not contained in the "closure of (A intersect B)".

I was considering possible cases:

(a) x is a limit point of A and an interior point of B, but then x should be in the closure of (A intersect B)... I could/hope I'm wrong.

(b) x is a limit point of both A and B. This seemed to go nowhere.

(c) x is in the interior of A and in the interior of B. I am pretty sure this will go nowhere.

When I think of A and B, I think of Venn diagrams or the real line. I guess I should be more creative, huh?

Any help will be appreciated. Thanks!!!!

Answer 1

what if A = (0,1) and B = (1,2)......A intersect B = null set while [0,1] intersect [1,2] = {1}

Answer 2

Even better: The rationals and irrationals.

intersection of closures: real line.

closure of intersection: empty set.

Steve