Boundary of Union is Subset of Union of Boundaries

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

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

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

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

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

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

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

Thus