Boundary of Boundary of Subset of Indiscrete Space

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Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Let $H \subseteq S$.


Then:

$\partial \left({\partial H}\right) = \varnothing$

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


Proof

From Boundary of Subset of Indiscrete Space, either $\partial H = S$ or $\partial H = \varnothing$, depending on whether $H = \varnothing$ or $H = S$ or not.

From Open and Closed Sets in Topological Space, $\varnothing$ and $S$ are both closed and open in $T$.

So from Set Clopen iff Boundary is Empty $\delta H = \varnothing$.

$\blacksquare$


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