All spaces are assumed Hausdorff. We call a topological space compact if every open cover has a finite subcover. We call it Lindelöf if every open cover has a countable subcover, and hereditarily Lindelöf if moreover every subspace is Lindelöf.
It is obvious that every compact space is Lindelöf. We have that every closed subset of a compact space is compact, i.e. it is Lindelöf and 'so much more'. This leads to the natural question, is every open set Lindelöf, too? (this would suffice to make the space hereditarily Lindelöf; see for example here).