Summary: We show first that every topology \(\tau\) has a minimum Alexandroff topology expansion \(\tau^{A}\) and investigate such expansion topologies. Then, we lift the Ginsburg structure theorem for homogeneous finite spaces to the class of homogeneous partition spaces which includes the class of homogeneous locally finite spaces. Finally, we introduce the class of upper bounded (lower bounded) \(T_{0}\) Alexandroff spaces which properlycontains the class of Artinian (Noetherian) \(T_{0}\) Alexandroff spaces and prove that a \(T_{0}\) Alexandroff space is compact if and only if it is both lower bounded and has a finite set of minimal elements in the specialization order.