Boundary of Boundary is Contained in Boundary

Theorem

Let $T$ be a topological space, and let $H \subseteq T$.

Then:

$\partial \left({\partial H}\right) \subseteq \partial H$

where $\partial H$ is the boundary of $H$.

That is, the boundary of the boundary of $H$ is contained in the boundary of $H$.

Proof

Let $B = \partial H$.

From Boundary of Set is Closed we have that $B$ is closed in $T$.

Let $B^-$ denote the closure of $B$.

From Boundary is Intersection of Closure with Closure of Complement:

$\partial B = B^- \cap \left({T \setminus B}\right)^-$

and so from Intersection is Subset:

$\partial B \subseteq B^-$

But from Closed Set Equals its Closure:

$B = B^-$

and so:

$\partial B \subseteq B$

That is:

$\partial \left({\partial H}\right) \subseteq \partial H$

$\blacksquare$

Sources

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 1$: Closures and Interiors