The concepts of semicompactness, countable semicompactness, and the semi-Lindelöf property are introduced in L -topological spaces, where L is a complete de Morgan algebra. They are defined by means of semiopen L -sets and their inequalities. They do not rely on the structure of basis lattice L and no distributivity in L is required. They can also be characterized by semiclosed L -sets and their inequalities. When L is a completely distributive de Morgan algebra, their many characterizations are presented.