Some Category Properties Of Z-Connected Set System

In this paper, for a Z-connected set system, the concept of Z-connected algebraic poset is introduced. Cartesian closedness of category of Z-connected poset is discussed. We prove that the category of Z-connected algebraic posets is dually equivalent to a full subcategory of the category of strongly algebraic lattices. Finally, using only the axiom DCω of ω dependent choices, we study that the Z-connected continuous poset A is completely regular, which is equipped with a countable base, the space A with the Z-connected Lawson topology is metrizable space.