Boundary of Boundary is not necessarily Equal to Boundary
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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.
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