Boundary of Cartesian Product of Subsets

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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 $H$.


Proof

\(\ds \map \partial {H \times K}\) \(=\) \(\ds \map \cl {H \times K} \setminus \Int {H \times K}\) Definition of Boundary (Topology)
\(\ds \) \(=\) \(\ds \paren {\map \cl H \times \map \cl K} \setminus \Int {H \times K}\) Closure of Cartesian Product is Product of Closures
\(\ds \) \(=\) \(\ds \paren {\map \cl H \times \map \cl K} \setminus \paren {\Int H \times \Int K}\) Interior of Cartesian Product is Product of Interiors
\(\ds \) \(=\) \(\ds \paren {\map \cl H \times \paren {\map \cl K \setminus \Int K} } \cup \paren {\paren {\map \cl H \setminus \Int H} \times \map \cl K}\) Set Difference of Cartesian Products
\(\ds \) \(=\) \(\ds \paren {\map \partial H \times \map \cl K} \cup \paren {\map \cl H \times \map \partial K}\) Definition of Boundary (Topology)

$\blacksquare$


Sources

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