Boundary of Subset of Discrete Space is Null
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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: $4$