Boundary of Subset of Discrete Space is Null

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Theorem

Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Let $A \subseteq S$.


Then:

$\partial A = \varnothing$

where:

$\partial A$ is the boundary of $A$ in $T$.


Proof

Let $A \subseteq S$.

Then from Set in Discrete Topology is Clopen it follows that $A$ is both open and closed in $T$.

The result follows from Set Clopen iff Boundary is Empty.

$\blacksquare$


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