Finite Quasi-uniform Extensions. II

The lattice of finite extensions of quasi-uniformities for prescribed topologies are examined. An example is presented which shows that in the finite and strict case there can occur that there exists no coarsest compatible extension. It is also verified that inf ≠ ⋒ in the lattice of extensions. In the strict case there exists a coarsest compatible extension if and only if there exists a coarsest extension for X ∪ {p} for every p ∈ Y - X. It is shown that certain special sup-distributive lattices can be represented by the lattice of extensions. For example every finite distributive lattice is isomorphic to the lattice of extensions for a suitable system.