Boundary of Boundary is not necessarily Equal to Boundary

Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

Let $\partial H$ denote the boundary of $H$.

While it is true that:

$\map \partial {\partial H} \subseteq \partial H$

it is not necessarily the case that:

$\map \partial {\partial H} = \partial H$

Proof

From Boundary of Boundary is Contained in Boundary, we have that:

$\map \partial {\partial H} \subseteq \partial H$

It remains to be proved that the equality does not always hold.

Proof by Counterexample:

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

Let $H \subseteq S$ such that $H \ne \O$ and $H \ne S$.

From Boundary of Subset of Indiscrete Space:

$\partial H = S$

From Boundary of Boundary of Subset of Indiscrete Space:

$\map \partial {\partial H} = \O$

The result follows.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors