T0-Closure Operators and Pre-Orders

Abstract

It is well known that the lattice of closed subsets of any topological space is isomorphic to that of a T₀-topological space. This result is extended to lattices of closed subsets with respect to arbitrary closure operator on a set. Also, we establish a one-to-one correspondence between closure operators which are both algebraic and topological on a given set X and pre-orders on X and prove that this correspondence induces a one-to-one correspondence between topological algebraic T₀-closure operators on X and partial orders on X.