Cantor Space is not Extremally Disconnected

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Theorem

Let $T = \left({\mathcal C, \tau_d}\right)$ be the Cantor space.


Then $T$ is not extremally disconnected.


Proof

From Cantor Space satisfies all Separation Axioms we have that $T$ 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 $T$.

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 $C_2$ in $T$.

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

The result follows from Extremally Disconnected by Disjoint Open Sets.

$\blacksquare$


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