Closure lattices

Closure spaces have been previously investigated by Paul Edelman and Robert Jami-son as \convex geometries". Consequently, a number of the results given here duplicate theirs. However, we employ a slightly diierent, but equivalent, deening axiom which gives a new avor to our presentation. The major contribution is the deenition 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, though non-modular, properties. Investigation of this lattice becomes our primary focus.