Topogenous and Nearness Structures on Categories

Quasi-uniform structures determined by closure operators

“…The categorical topogenous order that is defined above allows to obtain a nice relationship between closure and interior operators on a category in the sense that many concepts and definitions that have been studied separately for categorical closure and interior operators can be shown to be exactly the same using topogenous orders (see for example the notion of -strict subobject defined and used in third section of this paper). The next result that we recall from ( [9]) exhibits the clear relationship between closure, interior and topogenous order in a category while it is also known from the same paper that a topogenous order on C is basically equivalent to a neighbourhood operator ( [10]). Proposition 2.1.…”

“…For the basic facts on categorical closure and interior operators we refer to [4,5] and [14]. Concerning the categorical topogenous, quasi-uniform and syntopogenous structures, we use [12,8,9]. Throughout the paper, we consider a category C supplied with a proper (E, M)-factorization system for morphisms.…”

“…We show that there is a subconglomerate of the conglomerate of all quasi-uniform structures on C which is order isomorphic to the conglomerate of all idempotent closure operators. With the help of categorical topogenous structures ( [9]), we demonstrate that for any idempotent interior operator i (closure c) on C, there is at least a transitive quasi-uniformity which generates i (c). A condition under which a topogenous order is compatible with a transitive quasiuniformity is found.…”

Neighbourhood Operators: Additivity, Idempotency and Convergence

“…To avoid repetition, we shall only cite examples that are recent and refer the reader to [5][6][7]21,23,24] for further illustrations. 4.…”

“…5. A topogenous order on a set is interpolative if and only if the neighbourhood operator associated to it is idempotent [21]. 6.…”