Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 26, 1273 - 1287 The Categories of Textures and Topological Spaces S ¸enol Dost Hacettepe University, Department of Secondary Science and Mathematics Education, 06800, Beytepe, Ankara, Turkey dost@hacettepe.edu.tr Abstract In this study, mainly, we introduce the category Tex of textures and texturally continuous functions. Relations with some other cat- egories, particulary dfTex, fTex, are pointed out. It is shown that the categories CTop of C-spaces and a full subcategory of Sober of sober spaces are, respectively, isomorphic to Tex and STex of simple textures. Some properties discussed including the existence of product, coproduct, equalizer coequalizer. Mathematics Subject Classification: 54B05, 54B10, 54C60, 54C45 Keywords: Texture, Adjoint functor, Sober space, Isomorphism functor, Product, Coproduct, Equalizer, Coequalizer, Complete, Cocomplete 1 Introduction The theory of texture spaces was introduced by L.M. Brown in 1992 under the name “fuzzy structure”, and results on this topic appear in several papers including [3-7,11,12]. Texture space: [4] Let S be a set. A texturing is a point-separating, com- plete, completely distributive lattice with respect to inclusion, which contains S and ∅, and for which arbitrary meets coincide with intersections, and finite joins with unions. If S is a texturing of S the pair (S, S) is called a texture space or simplify a texture for short. For s ∈ S the sets P s =  {A ∈ S | s ∈ A} and Q s =  {P u | u ∈ S, s / ∈ P u } are called respectively, the p-sets and q-sets of (S, S). In a texture, arbitrary joins need not coincide with unions, and clearly this will be so if and only if P s  Q s for all s ∈ S . In this case (S, S) is said to be plain.