A new form of α-compactness is introduced in L-topological spaces by α-open L-sets and their inequality where L is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice L. It can also be characterized by means of α-closed L-sets and their inequality. When L is a completely distributive de Morgan algebra, its many characterizations are presented and the relations between it and the other types of compactness are discussed. Countable α-compactness and the α-Lindelöf property are also researched.
In this paper we introduce and study the concepts of HC-closed set and HClimit (HC-cluster) points of L-nets and L-ideals using the notion of almost N-compact remoted neighbourhoods in L-topological spaces. Then we introduce and study the concept of HL-continuous mappings. Several characterizations based on HC-closed sets and the HC-convergence theory of L-nets and L-ideals are presented for HL-continuous mappings.