On Subset Families That Form a Continuous Lattice

It is well known that continuous lattices and algebraic lattices can be respectively represented by the family of all fixed points of the projection operator and the closure operator preserving sups of directed sets on the power set of a set X. Similar to the algebraic ⋂-structure as the concrete representation of algebraic lattices, can we have a the concrete representation of continuous lattices by families of sets? We give a positive answer to this in this paper. Also, as a special type of continuous lattice, a concrete representation of completely distributive complete lattices by families of sets is obtained.