Open Unit Interval on Rational Number Space is Bounded but not Compact

Theorem
Let $\left({\Q, \tau_d}\right)$ be the rational number space under the Euclidean topology $\tau_d$.

Let $I := \left({0 \,.\,.\, 1}\right)$ be the open unit interval.

Then:
 * $\left({0 \,.\,.\, 1}\right) \cap \Q$ is totally bounded but not compact

where $\left({0 \,.\,.\, 1}\right)$ is the open unit interval.

Proof
From the Heine-Borel Theorem on a metric space, $\left({0 \,.\,.\, 1}\right) \cap \Q$ is compact iff it is both totally bounded and complete.

From Rational Number Space is not Complete Metric Space it follows that $\left({0 \,.\,.\, 1}\right) \cap \Q$ is not compact.