On reflexive closed set lattices

For a topological space X, let S(X) denote the set of all closed subsets in X, and let C(X) denote the set of all continuous maps f : X -> X. A family A subset of S(X) is called reflexive if there exists C subset of C(X) such that A = {A is an element of S(X) : f(A) subset of A for every f is an element of C}. Every reflexive family of closed sets in space X forms a sub complete lattice of the lattice of all closed sets in X. In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed sets to be reflexive are obtained.