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

Textural approach to generalized rough sets based on relations

“…respectively (for topological arguments on textures see [1,[4][5][6][7][8]10,11,17]). Product of textures can be defined in a natural way [3].…”

“…Hence, many properties of Hutton algebras (fuzzy lattices) can be discussed in terms of textures [3]. Ditopologies (dichotomous topologies) on textures unify the fuzzy topologies, topologies and bitopologies in a non-complemented setting by means of duality in the textural concepts [1,[4][5][6][7][8]10,11,17]. Direlations are defined between textures as the pairs of elements of the product of textures, that is if ðU; UÞ and ðT; TÞ are textures, then ðr; RÞ is a direlation from U to T where r; R 2 PðUÞ T [4].…”

Dual closure operators and their applications

“…This notion, originally designed to help characterize epimorphisms in subcategories of topological spaces and to determine whether such subcategories are cowell-powered (so that every object X allows for only a set of nonisomorphic epimorphisms with domain X), has enjoyed considerable attention; see in particular the monographs [23,8]. Its applications range from topology to algebra and theoretical computer science; see, for example, [22,24,6,14,19,20]. What is the categorically dual notion of closure operator?…”

“…Starting with [51], in recent years several authors have investigated categorical interior operators, with the formation of the interior of a subspace of a topological space providing the role model; see [9,10,31,19,37]. While in Section 6 of this paper we make precise in which sense this notion is an order-dualization of the notion of closure operator, it certainly does not address the quest for the categorical dual of the notion of closure operator.…”

Quotient structure of interior-closure texture spaces

“…Closure spaces were introduced by Cech [9] and closure operators have been used intensively in some branches of mathematics such as Topology and Algebra. In [15], a discussion was given on closure operators in the sense of Cech on texture spaces. Besides, interior-closure operators on texture spaces in the sense of Dikranjan-Giuli which give more suitable environments for different areas were discussed in [16].…”