Journal of Advanced Studies in Topology ISSN: 2090-388X online Vol. 2, No. 1, 2011, 1–8 c  2011 Modern Science Publishers www.m-sciences.com RESEARCH ARTICLE Some Decompositions of Bicontinuous Difunction S ¸enol Dost ∗ Hacettepe University, Department of Secondary Science and Mathematics Education, 06800 Beytepe, Ankara, Turkey. (Received: 8 November 2010 , Accepted: 20 November 2010 ) In this paper, the counterparts of A-continuity and A-cocontinuity for difunctions are introduced based on A-set and co- A-set in ditopological texture spaces. On the other hand, locally-continuity and locally-cocontinuity for difunctions are given based on locally-closed set and locally-coclosed set. Finally, the some decompositions of bicontinuous difunctions are presented. Keywords: Texture spaces; A-set; A-bicontinuous difunction; locally closed sets; locally bicontinuous difunction; decomposition. AMS Subject Classification: 54A05, 54C10. 1. Introduction In general topology, over the last year that different types of generalized open sets are given. And accordingly, continuities in weak sense have been investigated in detail. Note that for each weak form of continuity, there is another weak form of continuity such that both them imply continuity. The mentioned generalized structures are used as a tool to solve some problems. For example, locally closed sets which are given by Bourbaki [1], plays an important role the concept of simple extensions [2]. On the other hand, pre-continuous maps [3] are important in functional analysis in the context of open mapping and closed graph theorems [4]. Besides, these generalized structures lead to solve the problem of finding the continuity dual of some generalized continuous functions in order to have a decomposition of continuity. In (ditopological) texture spaces theory, in order find effective properties of bicontinuity many weak forms of bicontinuity may be useful. Actually, until now, some generalized open (- closed) sets and generalized bicontinuous difunctions are given in ditopological texture spaces [5–8]. In this paper, some new concepts of generalized difunctions are studied base on (co)-A-sets and locally-(co)closed sets, and some of their properties and the relationship with the other generalized stuructured are presented. Finally, we give four different decompositions of bicontiuous difunction. Textures and ditopological texture spaces were first introduced by L. M. Brown as a point-based setting for the study fuzzy topology. This section concludes with some basic definitions from the theory, and the reader is referred to [9–14] for more background material. If S is a set, a texturing S of S is a point-separating, complete, completely distributive lattice containing S and ∅, and for which arbitrary meets coincide with intersections, and finite joins with unions. The pair (S, S) is then called a texture space or shortly, texture. † Email: dost@hacettepe.edu.tr