On the Representations of an Abstract Lattice as the Family of Closed Sets of a Topological Space

Characterizing topologies by functions

“…If L+(X) has its idempotents isomorphic to those of L+(Y) then C(X) is lattice isomorphic to C(Y). Since both spaces are Tn spaces, by [3, Theorem 2.1], Xis homeomorphic to F. The topology of X can be explicitly recovered from C(X) as in [1].…”

Stone compactification of additive generalized-algebraic lattices

“…C is the set of all closed subsets. D. Drake, W. J. Thron and S. Papert considered C as a complete lattice (C, ∪, ∩, ∅, X)( [11,16]). But unfortunately the correspondence between complete lattices and T 0 -topological spaces is not one-to-one.…”

“…To solve the problem, Deng also investigated generalized continuous lattices on the basis of [1,11,15,16]. He introduced the notions of maximal systems of subsets, additivity property, homomorphisms, direct sums, lower sublattices in [5,6,9,10].…”

Lower homomorphisms on additive generalized algebraic lattices

“…On the other hand, suppose (X, C) is a co-topological space and C the set of all closed subsets of a topological space on X. D. Drake, W. J. Thron, S. Papert considered C as a complete lattice (C, ∪, ∩, ∅, X)( [11,16]). But unfortunately the correspondence between complete lattices and T 0 -topological spaces is not one-to-one.…”

“…To solve the problem, on the basis of [1,11,15,16], Deng also investigated generalized continuous lattices. He introduced the notions of the maximal system of subsets, additivity property, and homomorphisms in [5,6,7,10].…”