Heine-Borel Theorem states that if a set has an open cover and if we can find a finite subcover from that open cover that covers the set, the set would be compact. I got this question while I was trying to prove that Heine-Borel property will imply that the set is closed.
Is there any restriction on open sets that are the contained in the open cover? If not, can't I just make one of the open sets to be a set of real numbers? Then, the set would always have a finite subcover, that is, the set of all real numbers that covers the set. Hence, it can cover $(0,1)$ or any other open intervals but still $(0,1)$ is not closed.