every -compact set is Lindelöf
Every -compact (http://planetmath.org/SigmaCompact) set is Lindelöf (every open cover has a countable subcover).
Let be a -compact. Let be an open cover of . Since is -compact, it is the union of countable many compact sets,
with compact. Consider the cover of the set . This cover is well defined, it is not empty and covers : for each there is at least one of the open sets such that .
Since is compact, the cover has a finite subcover. Then
That is, the set is a countable subcover of that covers . ∎