Formula for the Boundary of an intersection

Question

I'm wondering if there's a formula for $\partial(A\cap B)$. This isn't homework, just wondering.

Problems arise because in general $\overline{A\cap B}\ne \overline A\cap \overline B$ — Hagen von Eitzen, Oct 06 '17 at 17:35
This question is likely to be closed for lack of context. Can you describe the sort of problems that started you wondering about this? — Jacob Manaker, Jun 26 '21 at 04:40

William Elliot · Answer 1 · 2019-09-30T22:20:16.527

2

For what it is worth.

$\partial A \cup \partial B = \partial (A\cup B) \cup \partial (A\cap B) \cup (\partial A \cap \partial B)$

edited Sep 30 '19 at 22:20

answered Oct 07 '17 at 05:57

William Elliot

17,598

score 0 · Answer 2 · answered Apr 18 '20 at 22:51

There is one indeed, but it's a bit clunky unless $\partial{A} \cap \partial{B}=\varnothing$ and $A$ and $B$ are closed. The case for abritrary open sets in an arbitrary space appears here.

A similar identity holds for the boundary of a Cartesian product, whose striking similarity to a product rule is discussed here.

Formula for the Boundary of an intersection

2 Answers2