In this paper, the notions of SPN-compactness, countable SPNcompactness and the SPN-Lindelöf property are introduced in L-topological spaces by means of strongly preclosed L-sets. In an L-space, an Lset having the SPN-Lindelöf property is SPN-compact if and only if it is countably SPN-compact. (Countable) SPN-compactness implies (countable) N-compactness, the SPN-Lindelöf property implies the N-Lindelöf property, but each inverse is not true. Every L-set with finite support is SPN-compact. The intersection of an (a countable) SPN-compact L-set and a strongly preclosed L-set is (countably) SPNcompact. The strong preirresolute image of an (a countable) SPNcompact L-set is (countably) SPN-compact. Moreover SPN-compactness can be characterized by nets.