Sequentially compact subsets and monotone functions: An application to fuzzy theory

Abstract

Let (X,<,τO) be a first countable compact linearly ordered topological space. If (Y,D) is a uniform sequentially compact linearly ordered space with weight less than the splitting number s, then we characterize the sequentially compact subsets of the space M(X,Y) of all monotone functions from X into Y endowed with the topology of the uniform convergence induced by the uniformity D. In particular, our results are applied to identify the compact subsets of M([0,1],Y) for a wide class of linearly ordered topological spaces, including Y=R. This allows us to provide a characterization of the compact subsets of an extended version of the fuzzy number space (with the supremum metric) where the reals are replaced by certain linearly ordered topological spaces, which corrects some characterizations which appear in the literature. Since fuzzy analysis is based on the notion of fuzzy number just as much as classical analysis is based on the concept of real number, our results open new possibilities of research in this field