Question: What condition of the Heine-Borel Theorem does this not meet?

Consider $\mathbb Q$ with metric $d(p,q) = |p-q|$, and
$$ E = \{q : 2 < p^2 < 3\}. $$

We can show that $E$ is closed and bounded and $E$ is not compact.

$E$ is a subset of $\mathbb R^1$ and euclidean space with the appropriate metric.


To apply Heine-Borel, you need your space to be complete. But $\mathbb Q$ isn't.

Note that, as a subset of $\mathbb R$, your set $E$ is not closed.

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