Boundary of Intersection is Subset of Union of Boundaries

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

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

Then:
 * $\map \partial {A \cap B} \subseteq \partial A \cup \partial B$

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

Proof
By Intersection is Subset:
 * $A \cap B \subseteq A \land A \cap B \subseteq B$

Then by Topological Closure of Subset is Subset of Topological Closure:
 * $\paren {A \cap B}^- \subseteq A^- \land \paren {A \cap B}^- \subseteq B^-$

Hence by Boundary is Intersection of Closure with Closure of Complement:
 * $\paren {A \cap B}^- \cap \paren {\relcomp S A}^- \subseteq \partial A \land \paren {A \cap B}^- \cap \paren {\relcomp S B}^- \subseteq \partial B$

Thus