Boundary of Cartesian Product of Subsets

Theorem
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.

Let $H \subseteq T_1$ and $K \subseteq T_2$.

Then:
 * $\map \partial {H \times K} = \paren {\map \partial H \times \map \cl K} \cup \paren {\map \cl H \times \map \partial K}$

where:
 * $\map \cl H$, for example, denotes the closure of $H$.
 * $\map \partial H$, for example, denotes the boundary of $H$.