In this paper, \(\mathcal {P}(L)\) denotes the set of prime ideals, Min(L) denotes the set of minimal prime ideals and Max(L) denotes the set of maximal ideals of a meet semilattice L. The set \(\mathcal {P}(L)\) can be endowed with the well-known topology called spectral topology. \(\mathcal {P}(L)\) can also be endowed with another subtopology of the spectral topology called D-topology. In this paper, we prove that if a meet semilattice \(L \in \mathbb {P}_{MIP} \cap \mathbb {P}_{MFP}\) then the spectral topology and D-topology coincide on \(\mathcal {P}(L)\) , Min(L) and Max(L) if and only if L is a complemented, Stone and pm-meet semilattice, that is every prime ideal is contained in a unique maximal ideal respectively.
