Textures and C-spaces

In this paper we point out that textures, simple textures and plain textures are, respectively, isomorphic to C-spaces, sober C-spaces and partially ordered sets. Using the isomorphism between textures and C-spaces, we show that subtextures, quotient textures, sums and products of textures can be identified as subspaces, quotient spaces, sums and products of C-spaces. Furthermore, it is shown that textures, except discrete textures, do not satisfy any separation axioms (T1,…,T4) other than T0, and so they are not pseudometrizable. Then we also give four useful characterizations of the complete distributivity condition of textures.