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.