Rational Number Space is not Extremally Disconnected

Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.

Then $\struct {\Q, \tau_d}$ does not form an extremally disconnected space.

Proof
We have that $\openint 0 1$ is open in $\struct {\R, \tau_d}$.

Thus $\openint 0 1 \cap \Q$ is open in $\struct {\Q, \tau_d}$.

We show that $0, 1$ are limit point of $\openint 0 1 \cap \Q$.

For any $\epsilon > 0$:


 * $\openint {-\epsilon} \epsilon \cap \Q \cap \openint 0 1 = \openint 0 \epsilon \cap \Q$

From Between two Real Numbers exists Rational Number:


 * $\openint 0 \epsilon \cap \Q \ne \O$

Similarly:


 * $\openint {1 - \epsilon} 1 \cap \Q \ne \O$

Thus $0, 1$ are limit point of $\openint 0 1 \cap \Q$.

Hence $0, 1 \in \paren {\openint 0 1 \cap \Q}'$.

Thus $\closedint 0 1 \cap \Q \subseteq \paren {\openint 0 1 \cap \Q}^-$.

Now we show that for any $x \notin \closedint 0 1$, $x$ is not a limit point of $\openint 0 1 \cap \Q$.

Suppose $x < 0$.

Then $\dfrac x 2 < 0$.

Thus $\openint {x + \dfrac x 2} {x - \dfrac x 2} \cap \Q \cap \openint 0 1 = \O$.

Similarly for $y > 1$:


 * $\openint {y - \dfrac {y - 1} 2} {y + \dfrac {y - 1} 2} \cap \Q \cap \openint 0 1 = \O$

Thus $\map \complement {\closedint 0 1 \cap \Q} \subseteq \map \complement {\paren {\openint 0 1 \cap \Q}^-}$.

Hence we have $\closedint 0 1 \cap \Q = \paren {\openint 0 1 \cap \Q}^-$.

From Between two Real Numbers exists Rational Number:


 * $\openint {-\epsilon} \epsilon \cap \Q \cap \map \complement {\closedint 0 1} = \openint {-\epsilon} 0 \cap \Q \ne \O$

Any neighborhood of $0$ must intersect $\complement {\closedint 0 1}$.

So $\closedint 0 1 \cap \Q$ is not open in $\struct {\Q, \tau_d}$.

There exists some open set in $\struct {\Q, \tau}$ where its closure is not open.

Thus the result follow from definition of extremally disconnected space.