The concepts of $\gamma$-compactness, countable $\gamma$-compactness, the $\gamma$-Lindelöf property are introduced in $L$-topological spaces by means of $\gamma$-open $L$-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. .@filename: kjom35(1)-03.pdf