Existence of lattice-valued uniformly continuous mappings

Abstract

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/sup X/, D) be an L-fuzzy uniform space, f /spl isin/ D, A, B /spl isin/ L/sup X/ such that f (A) /spl les/ B. Then there exists an L-fuzzy uniformly continuous mapping F/sup /spl rarr// : (L/sup X/, D) /spl rarr/ I(L) such that A /spl les/ F/sup /spl larr//(L/sub 1/') /spl les/ F/sup /spl larr//(R/sub 0/) /spl les/ B. In the outline of its proof, Hutton affirmed that one could find {h/sub r/ : r > 0} /spl sub/ D and {A/sub s/ : s /spl isin/ R} /spl sub/ L/sup X/ such that h/sub r/(A/sub s/) /spl les/ A/sub 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 /spl rarr/ 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/sup X/, L/sup Y/, etc.