The definition means that if we can cover a compact set withanyparticular collection ofopen setsthen there is a finite subcollection of those very sets which also covers the compact set.

It does not say that if a set has a finite open covering then the set is compact.

Rather, for a compact set, from any open covering of that set we can find a finite subcovering.

I have tried to put that two different ways. I hope that helps.