Existence of lattice-valued uniformly continuous mappings

In L-fuzzy topology, the theorem of existence of uniformly continuous mappings is very essential for the theory of uniform spaces and theory of metric spaces. In fact, the main and basic theorem "An L-fuzzy topological space is uniformizable if and only if it is completely regular" is just based on the theorem of existence of uniformly continuous mappings. B. Hutton (1977) introduced the theorem of uniformly continuous mappings in L-fuzzy topology as follows: Let (L ^X, D) be an L-fuzzy uniform space, f ∈ D, A, B ∈ L ^X such that f (A) ⩽ B. Then there exists an L-fuzzy uniformly continuous mapping F^→ : (L^X, D) → I(L) such that A ⩽ F^←(L₁\´) ⩽ F ^←(R₀) ⩽ B. In the outline of its proof, Hutton affirmed that one could find {h_r : r > 0} ⊂ D and {A_s : s ∈ R} ⊂ L^X such that h_r (A_s) ⩽ A_s-r. (*) This skeleton of a proof was widely accepted later but without concrete verifications. Some authors tried to complete this proof, such as Guo-Jun Wang (1988), but the efforts were not successful, and inequality (*) could not be fulfilled. A complete and concrete proof for the existence of lattice-valued uniformly continuous mappings is given, the errors appeared in some past proofs are corrected; they seem to mean that the widely accepted skeleton of proof is not correct. In the sequel, unless particular declaration, L always stand for an F-lattice, i.e. a completely distributive lattice with an order-reversing involution, then \´ : L → L. For convenience, we call sets of ordinary mappings from ordinary non-empty sets X, Y and so on to an F-lattice L L-fuzzy spaces, denoted by L^X, L^Y, etc