Algebraic Topological Closure Operators

Abstract

For any closure operator c there is a T o -closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T o ) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T o -lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T o -lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.