A compact topology for a lattice
Arnold J. Insel · Proceedings of the American Mathematical Society · 1963
In this paper we shall study a compact intrinsic topology for a lattice and obtain a few relationships between this topology and certain well-known intrinsic topologies for lattices.We obtain as a result the fact that for a large class of lattices, compactness of the order topology implies that our compact topology and the order topology coincide.Let A be a lattice and {x"}, a net in A. We define the limit inferior, ■L*{x0} = V0Af,S0 Xb, and the limit superior, A*{xa} = A"V¡,;>aX¡).Then, provided they exist, A*{x"} =A*{xa}.If A*{xa} =L*{xa) =x, we say that the net {x"} order converges to x.Let C be a subset of L. C is said to be order closed iff no net in C order converges to a point outside of C. The collection of order closed sets comprises the closed sets for a topology for A. We call this topology the order topology