example of non compact set for rationals

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there is an example from lectures which i do not understand.
given a closed set [0,1] , for the below set K, how is there not a finite subcover on K?

why is the set K=[0,1] $\bigcap \mathbb{Q}$ not compact?

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1 Answer

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Obverse that the cover $$[0,1]\cap \mathbb{Q} \subseteq (-1,\frac{\sqrt{2}}{2}) \cup \bigcup_{n \in \mathbb{N}}(\frac{\sqrt{2}}{2}+\frac{1}{n},2)$$

has no finite subcover.

Or you can use Heine-Borel Theorem.

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