The aim of this paper is to introduce a new concept of $\Lambda$ dense set and to study its topological structure. It can be shown that the collection of all this $\Lambda$ - dense set forms a supra topological space if $\phi$ is introduced. Also the concept of minimal $\Lambda$ dense set is introduced and some of its applications are shown. Lastly introducing the concept of $\Lambda$ sub maximal space various important properties of minimal $\Lambda$ dense sets are studied in a $\Lambda$- sub maximal space.