Boundary of Subset of Indiscrete Space

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Theorem

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

Let $\O \subsetneq H \subsetneq S$ (that is, let $H$ be a non-empty proper subset of $T$).


Then:

$\partial H = S$

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


If $H = \O$ or $H = S$ then $\partial H = \O$.


Proof

From Closure of Subset of Indiscrete Space, $H^- = S$, where $H^-$ denotes set closure.

From Interior of Subset of Indiscrete Space, $H^\circ = \O$, where $H^\circ$ denotes set interior.

By definition:

$\delta H = H^- \setminus H^\circ = S \setminus \O = S$


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:

$\delta H = \O$

$\blacksquare$


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