Abstract
The concepts of P-compactness, countable P-compactness, the P-Lindelöf property are introduced in \(L\)-topological spaces by means of preopen \(L\) -sets and their inequalities when \(L\) is a complete DeMorgan algebra. These definitions do not rely on the structure of the basis lattice \(L\) and no distributivity in \(L\) is required. They can also be characterized by means of preclosed L-sets and their inequalities. Their properties are researched. Further when \(L\) is a completely distributive DeMorgan algebra, their many characterizations are presented.
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