Abstract
In this paper, by means of semi-pre-open strong βa-cover and semi-pre-open βa-cover, the concept of strong-III Nβ-compactness is introduced for an arbitrary L-subset. The strong-III Nβ-compactness is different from SP-compactness. But when L = [0; 1], strong-III Nβ-compactness is equivalent to SP-compactness.