Interior and closure operators on texture spaces—I: Basic concepts and C˘ech closure operators

This paper is the first of a series of three papers on the theory of interior and closure operators. Here, the theory is discussed from the textural point of view. First, the interior and closure operators on texture spaces are defined and some basic properties are given in terms of neighbourhoods and coneigbourhoods. Then the category dfIC whose objects are interior–closure spaces and the morphisms are bicontinuous difunctions is shown to be topological over the ground category dfTex of textures and difunctions. Further, considering the closure operator on Hutton algebras (known as fuzzy lattices) in the sense of C˘ech, the category HutCl of Hutton closure spaces and continuous mappings is defined. Finally, the category cdfIC of complemented bicontinuous difunctions and complemented interior-closure texture spaces and the opposite category of HutCl are shown to be equivalent.