Boundary of Subset of Discrete Space is Null

Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

Let $A \subseteq S$.


Then:

$\partial A = \O$

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

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $4$
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