I have encountered this question: find a cover for $[1,2]$ in $\mathbb{R}$ that does not have a finite subcover.
Since $[1,2]$ is closed and bounded then it's compact. It then implies that a cover that does not have a finite subcover must not be open. I was thinking about $[1,1.5] \cup [1.5,2]$. However, it is not an infinite union of sets. Can it still be a cover?
Best Answer
Just cover it with one-element sets: $\{\{x\}, x\in[1,2]\}$. This is uncountable infinite cover and you cannot omit even a single one of them!