So if I use the definition of compactness that every open cover has a finite sub-cover, then as the unit ball is compact , there exists a finite subcover. But if I increase the radius of the ball, why does it still need to be compact. Intuitively speaking can't I just take very small sized and large number open sets in such a way that there is no finite sub-cover. I know that the answer to this question is no but I don't see why?
Can someone please throw some light?