Annals of Fuzzy Mathematics and Informatics Volume 4, No. 1, (July 2012), pp. 25- 48 ISSN 2093–9310 http://www.afmi.or.kr @FMI c  Kyung Moon Sa Co. http://www.kyungmoon.com T -syntopogenous spaces Khaled A. Hashem Received 14 June 2011; Accepted 16 September 2011 Abstract. In this paper, we introduce the concepts of T -syntopogenous spaces and investigate some of their properties, where T stands for any continuous triangular norm. Their definitions subsumes that of fuzzy syn- topogenous spaces due to A. K. Katsaras ( Fuzzy Sets and Systems 36 (1990)), as our Min-syntopogenous spaces. In particular, we study the con- tinuity of functions between T -syntopogenous spaces and the I -topological space associated with a T -syntopogenous space. Moreover, we describe the T -syntopogenous structures as fuzzy relations in (ordinary) power sets. 2010 AMS Classification: 62P30, 62A86 Keywords: Triangular norm, T -syntopogenous spaces, I -topological spaces Corresponding Author: Khaled A. Hashem (Khaledahashem@yahoo.com) 1. Introduction Katsaras and Petalas [5, 6, 7] introduced the fuzzy syntopogenous structures and studied the unified theory of Chang I -topologies [2] and Lowen Fuzzy uniformities [10]. In this manuscript, we introduce, for each continuous triangular norm T , a new structure of T -syntopogenous spaces that conforms well with Lowen I -topological spaces [9]. Our concept of T -syntopogenous structure generalizes, to arbitrary con- tinuous triangular norm T , the fuzzy syntopogenous structure of A. K. Katsaras [7], now becoming the special case corresponding to T = Min. Also we deduce the no- tion of syntopogenous maps and here we show that the class of all T -syntopogenous spaces together with syntopogenous maps as arrows forms a concrete category. The basic idea is to introduce a degree of divergence between fuzzy subsets, which is a real number in the unit interval I = [0, 1]. We proceed as follows: In Section 2, we state and supply some basic ideas and lemmas on the T -residuated implication and on the α-cuts of fuzzy subsets, which will be needed in the sequel. In the third section, we introduce our definition of T -topogenous order, and hence T -syntopogenous structure on a set, also we define the I -topology associated with a T -syntopogenous structure. We deduce the notions