An insertion theorem for continuous -valued functions and its consequences

Spaces in which open sets take values in the L-interval I(L) are investigated. We prove a theorem concerning the insertion of a continuous function with values in I(L) with L a complete lattice. We establish certain factorizations of the functors ωI(L) and ιI(L) with L a hypercontinuous lattice. The latter result and the insertion theorem provide a partial answer to a recent open question related to insertion of lattice-valued functions. As another application of the insertion theorem we establish a relation between complete regularity in the categories of L-topological and I(L)-topological spaces with L a frame.