Compactness of Lim-inf Topology

L {inf B; B ranges over subsets of L: B ∈ F}. One can prove the following proposition (1) Let L1, L2 be complete lattices. Suppose the relational structure of L1 = the relational structure of L2. Let X1 be a non empty subset of L1, X2 be a non empty subset of L2, F1 be a filter of 2 X1 ⊆ , and F2 be a filter of 2 X2 ⊆ . If F1 = F2, then lim inf F1 = lim inf F2. Let L be a non empty FR-structure. We say that L is lim-inf if and only if: (Def. 2) The topology of L = ξ(L). Let us note that every non empty FR-structure which is lim-inf is also topological space-like. One can check that every top-lattice which is trivial is also lim-inf. One can check that there exists a top-lattice which is lim-inf, continuous, and complete. We now state several propositions: (2) Let L1, L2 be non empty 1-sorted structures. Suppose the carrier of L1 = the carrier of L2. Let N1 be a net structure over L1. Then there exists a strict net structure N2 over L2 such that