Closure lattices

AbstractClosure spaces have been previously investigated by Paul Edelman and Robert Jamison as ‘convex geometries’. Consequently, a number of the results given here duplicate theirs. However, we employ a slightly different, but equivalent, defining axiom which gives a new flavor to our presentation.The major contribution is the definition of a partial order on all subsets, not just closed (or convex) subsets. It is shown that the subsets of a closure space, so ordered, form a lattice with regular, although non-modular, properties. Investigation of this lattice becomes our primary focus