Boundary of Boundary of Subset of Indiscrete Space
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Theorem
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $H \subseteq S$.
Then:
- $\map \partial {\partial H} = \O$
where $\partial H$ denotes the boundary of $H$.
Proof
From Boundary of Subset of Indiscrete Space, either $\partial H = S$ or $\partial H = \O$, depending on whether $H = \O$ or $H = S$ or not.
From Open and Closed Sets in Topological Space, $\O$ and $S$ are both closed and open in $T$.
So from Set Clopen iff Boundary is Empty:
- $\map \partial {\partial H} = \O$
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $6$