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
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:
 * $\left({A \cap B}\right)^- \subseteq A^- \land \left({A \cap B}\right)^- \subseteq B^-$

Hence by Boundary is Intersection of Closure with Closure of Complement:
 * $\left({A \cap B}\right)^- \cap \left({\complement_S \left({A}\right)}\right)^- \subseteq \partial A \land \left({A \cap B}\right)^- \cap \left({\complement_S \left({B}\right)}\right)^- \subseteq \partial B$

Thus