Fuzzy Sets and Systems 6 (1981) 197-204 North-Holland Publishing Company THE STON-E-CECH COMPACTIFICATION IN THE CATEGORY OF FUZZY TOPOLOGICAL SPACES Umberto CERRUTI lstituto di Geometria, Universitfi di Torino, Via Principe Amedeo ~, Torino, Italy Received August 1979 Revised February 1980 In this paper we obtaihl a reflective subcategory ~ of the category FIrS ~ff fuzzy topological spaces. The associated reflection /3 has properties similar to those of the "Stone-('ech" compactification ~ and, in effect, is an extension of it. We study relations between 18 and /3 in particular subcategories of FTS; /3 is completely determined in the case of fuzzy topological spaces topologically generated. Keywords: Fuzzy topology, Fuzzy compactness, Compactification theory. 1. Introduction and preliminary remarks In this paper our principal aim is to clarify - through an effective use of category theory-some concepts (like compactnessl in the theory of fuzzy topological spaces (f.t.s.), and to give a new powerful tool (a compactification theory) for the study of them. We use the definition of f.t.s, given in [1], and we call the resulting category FTS; so the defi,~ition given in [7] determines a (full) subcategory of FI'S that we call K-FTS. In ~-7] the author introduces the functor to: Top-, K-FI'S and proves that it is an embedding; of course, there is an other - obvious - embedding of TOP in FTS, by characteristic functions of open sets; we call it j. We prove that the category A-FTS, whose objects are the f.t.s.'s (X, MJ for which M is finer than j(L(,~))-where ~ is the functor defined in [7]-has the following properties: (a) The intersection A-FTS fq K-FTS consists exactly of topologically generated f.t.s.'s (Property 2.4(m)). (b) The functor ~ has ief~ and right adjoints, which are to and j, respectively (Property 2.5). (c) Many 'good' extensions of the concept of compactness, discussed x11 [9], are equivalent in A-FTS. So, we can speak simply of compact A-spaces. (b) and (c) suggest that A-FTS is particularly suitable for a compactification theory: actually the following holds: (d) The embedding of the subcategory HCA-FTS of Hausdorff compact A- spaces in FTS has a left adjoint /3 (Theorem 3.2). 0165-0114/81/0000-0000/$ 02.50 © North-Holland Publishing Company