# 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.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 1 - 3: \ 4$