Textures and C-spaces

摘要

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 (T_1,...,T_4) other than T_0, and so they are not pseudometrizable. Then we also give four useful characterizations of the complete distributivity condition of textures.