In this paper, a new definition of fuzzy compactness is presented in L-topological spaces when L is a complete DeMorgan algebra. This definition does not rely on the structure of the basis lattice L and no distributivity in L is required. The intersection of a fuzzy compact L-set and a closed L-set is fuzzy compact. The continuous image of a fuzzy compact L-set is fuzzy compact. If the set of all prime elements of L is order generating, then the Alexander Subbase Theorem is true. When L is a completely distributive DeMorgan algebra, it is equivalent to Lowen's fuzzy compactness, and in this case, its many characterizations are presented by means of open L-sets and closed L-sets.