Intrinsic topologies in topological lattices and semilattices

Jimmie Lawson
1973 Pacific Journal of Mathematics  
{L(x):xeL} and all sets {M{x):xeL}. If L does not have universal bounds, then a set K c L is closed if K Π [a, b] is closed in the interval topology of the sublattice [α, b] for all a, b with a ^ 6. (2) The order topology (0). A net {x a } in L is said to orderconverge to # if there exist a monotonic ascending net {t a } with a;sup t a (t a t a?) and a monotonic descending net {u a } with a? = inf u a {u a \ x) such that for all α, ί β ^ # α ^ w β . A subset A of L is ciosβd in the order
more » ... y if {x a } c A and x α order converges to x imply that xeA. Note that if x a order-converges to x, then for any cofinal subset of the domain directed set it remains true that x a order-converges to x. Hence the order topology may be defined equivalently by declaring a set U of L open if x e U and x a order-converges to x imply x a is residually in U. ( 3) The convex-order topology (CO). A subset ί7of L is a basic open set for the convex-order topology if (i) U is convex and (ii) if x a order-converges to x, x e U, then x a is residually in U. Again, the second condition is equivalent to U being open in the order topology. We now list some easily derived properties of these intrinsic topologies. PROPOSITION 3. If τ is an agreeable topology on a lattice L, then the CO topology is finer than τ. Proof. Since τ is locally convex, it suffices to show that if a convex set U is in τ, then it is open in the CO topology. Suppose that x a is a net that order-converges to x e U. Then there exist t a j x, u a I x such that for all a> t a <£ x a ^ u a . Since τ is agreeable, t a and u a are residually in U, and since U is convex x a is residually in U.
doi:10.2140/pjm.1973.44.593 fatcat:xaxdeohngja7xhi67l6uaxyccm