Given $K=[0,\infty)$, can I write $C=\{(-0.5,0.75),(0.25,\infty)\}$ or C={$R$} as open covers for $K$?
This a part of a confusion that I have regarding HBT (Heine-Borel Theorem). The HBT requires that sets are closed and bounded for it to have a finite subcover. If an open cover for $K$ can be written as above, this would mean that a subcover would also be finite. Also $R$ is open and can cover any subset of $R$. A subcover can be the cover itself. So $R$ can represent a finite subcover of $K$ (or any subset of $R$). But this does not fit in with HBT, because HBT requires $K$ to be bounded, which it is not. Can you please help me point out where exactly I am going wrong?
Best Answer
You are wrong when you claim that the Heine-Borel theorem requires that sets are closed and bounded for it to have a finite subcover. That theorem states that, if a subset of $\Bbb R^n$ is closed and bounded, then every cover has a finite subcover. It does not say that if a set is unbounded or not closed, then no open cover has a finite subcover.