# Boundary of Intersection is Subset of Union of Boundaries

## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A, B$ be subsets of $S$.

Then:

$\partial \left({A \cap B}\right) \subseteq \partial A \cup \partial B$

where $\partial A$ denotes the boundary of $A$.

## Proof

$A \cap B \subseteq A \land A \cap B \subseteq B$
$\left({A \cap B}\right)^- \subseteq A^- \land \left({A \cap B}\right)^- \subseteq B^-$
$\left({A \cap B}\right)^- \cap \left({\complement_S \left({A}\right)}\right)^- \subseteq \partial A \land \left({A \cap B}\right)^- \cap \left({\complement_S \left({B}\right)}\right)^- \subseteq \partial B$

Thus

 $\displaystyle \partial \left({A \cap B}\right)$ $=$ $\displaystyle \left({A \cap B}\right)^- \cap \left({\complement_S \left({A \cap B}\right)}\right)^-$ Boundary is Intersection of Closure with Closure of Complement $\displaystyle$ $=$ $\displaystyle \left({A \cap B}\right)^- \cap \left({\complement_S \left({A}\right) \cup \complement_S \left({B}\right)}\right)^-$ Complement of Intersection $\displaystyle$ $=$ $\displaystyle \left({A \cap B}\right)^- \cap \left({\left({\complement_S \left({A}\right)}\right)^- \cup \left({\complement_S \left({B}\right)}\right)^-}\right)$ Closure of Finite Union equals Union of Closures $\displaystyle$ $=$ $\displaystyle \left({A \cap B}\right)^- \cap \left({\complement_S \left({A}\right)}\right)^- \cup \left({A \cap B}\right)^- \cap \left({\complement_S \left({B}\right)}\right)^-$ Intersection Distributes over Union $\displaystyle$ $\subseteq$ $\displaystyle \partial A \cup \partial B$ Set Union Preserves Subsets

$\blacksquare$