Heine Borel Theorem explanation

Can somebody explain to me, how this works exactly?
I mean, in my opinion, I can find an open cover and a finite subcover of any set, for example, if i have the set A=]0, 2[ I would choose ]-1,3[ as my open cover and ]-0.5, 1.0[ consolidated with ]0.5, 2.5[ as my finite subcover, although ]0, 2[ is not compact..

Any help?

The Heine-Borel theorem states that a subset of R^n is closed AND bounded if and only if for every open cover, you can find a finite subcover.

For your set A, since it is open (and not closed) you can cover it by A itself, but you can't find a finite subcover that still covers A. You need A to be closed, i.e. it has to contain all of it's limit points.

You don't choose the finite subcover. Rather, the theorem says that a set, S, is compact if and only if it is closed and bounded.

So, once you've defined the open sets (hence, the topology), you should be able to find a finite collection of the open sets that still cover S.