Boundary of Intersection is Subset of Union of Boundaries

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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

\(\displaystyle \partial \left({A \cap B}\right)\) \(=\) \(\displaystyle \left({A \cap B}\right)^- \cap \left({\complement_S \left({A \cap B}\right)}\right)^-\) 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

$\blacksquare$


Sources