Ideal-valued topological structures

With L a complete lattice and M a continuous lattice, this paper demonstrates an adjunction between M -valued L-topological spaces (i.e. (L,M )-topological spaces) and Idl(M )-valued L-topological spaces where Idl(M ) is the complete lattice of all ideals of M . It is shown that the right adjoint functor provides a procedure of generating (L,M )-topologies from antitone families of (L,M )-topologies. This procedure is then applied to give an internal characterization of joins in the complete lattice of all (L,M )-topologies on a given set.