One dimensional topological lattices
Lee W. Anderson · Proceedings of the American Mathematical Society · 1959
The lattice theoretic terminology used in this paper is consistent with Birkhoff [2]. The topological terms can be found in [5] or [8] with the following exceptions. If X and Y are sets, X\Y denotes the relative compliment of Y with respect to X. If A is a subset of a topological space then A*, Aand FiA)=A*\A° denote the topological closure, interior and boundary of A. The symbol 0 denotes the empty set. We will agree that in usage of words common to topology and lattice theory, the topological meaning will take precedence. Thus to say that a subset A of a topological lattice is closed means A = A * and not A AA EA or A \/A EA. If L is a lattice and A is a subset of L we let