Cantor Set is Closed in Real Number Space

Theorem

Let $\CC$ be the Cantor set.

Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\CC$ is a closed subset of $\struct {\R, \tau_d}$.

Proof

By definition, the Cantor set is the complement of a union of open sets relative to the closed interval $\closedint 0 1$.

By the definition of a topology, that union is itself open in $\R$.

The closed interval $\closedint 0 1$ is itself the complement of a union of open sets $\openint \gets 0 \cup \openint 1 \to$.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set: $2$