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.