Cantor Space is not Extremally Disconnected

Theorem
Let $\left({\mathcal C, \tau_d}\right)$ be the Cantor set considered as a topological subspace of the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\mathcal C$ is not extremally disconnected.

Proof
From Cantor Space Satisfies All Separation Axioms we have that $\mathcal C$ is a $T_2$ (Hausdorff) space.

Consider the real number $\dfrac 1 4 = 0.020202 \ldots_3$.

We have that:
 * $C_1 := \mathcal C \cap \left[{0 .. \dfrac 1 4}\right)$
 * $C_2 := \mathcal C \cap \left({\dfrac 1 4 .. 1}\right]$

are disjoint sets both of which are open sets of $\mathcal C$.

However, as $\dfrac 1 4 \in \mathcal C$ we have that:
 * $\dfrac 1 4 \in C_1^-$

and
 * $\dfrac 1 4 \in C_2^-$

where $C_1^-$ and $C_2^-$ are the closures of $C_1$ and $C_2$.

Thus $C_1^- \cap C_2^- \ne \varnothing$.

The result follows from Extremally Disconnected by Disjoint Open Sets.