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Show directly that the set $[0, 1]\cap\mathbb{Q}$ does not have the Heine-Borel property. In other words, find an open cover of $[0, 1]\cap\mathbb{Q}$ that cannot be reduced to a finite subcover.

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    $\begingroup$ Hint: $1/\sqrt{2} \in [0, 1]$ but not in $[0, 1]\cap \mathbb{Q}$. $\endgroup$ – Jacky Chong Nov 6 '16 at 20:10
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Hint:

Consider the open cover

$$\left(-1, \frac{1}{\sqrt{2}}\right) \cup\left( \frac{1}{\sqrt{2}}+\frac{1}{n},2\right) $$

Suppose on the contrary you can find a finite subcover, use the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ to find an element in $\mathbb{Q} \cap [0,1]$ that is not covered.

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