Closure operators in semiuniform convergence spaces

Abstract

In this paper, the characterization of closed and strongly closed subobjects of an object in category of semiuniform convergence spaces is given and it is shown that they induce a notion of closure which enjoy the basic properties like idempotency,(weak) hereditariness, and productivity in the category of semiuniform convergence spaces. Furthermore, T1 semiuniform convergence spaces with respect to these two new closure operators are characterized.

T₁ Approach Spaces

“…Hence, there is no relation between T 1 (in the usual sense) and local T 1 . (iv) By Remark 2.12 (2) of [6], T 1 and local T 1 (i.e., T 1 at p for all p 2 X) axioms could be equivalent.…”

“…In 1991, Baran [2] introduced local T 1 separation property in order to de…ne the notion of strong closedness [2] in set-based topological category which forms closure operators in sense of Dikranjan and Giuli [14,15] in some well known topological categories Conv (category of convergence spaces and continuous maps) [6,18,23], Prord (category of preordered sets and order preserving maps) [7,15] and SUConv (category of semiuniform convergence spaces and uniformly continuous maps) [9,24]. Furthermore, Baran [2] generalized T 1 axiom of topology to topological category which is used to de…ne regular, completely regular and normal objects [4,5] in topological categories.…”

Tychonoff Objects in the Topological Category of Cauchy Spaces

“…The notions of "closedness" and "strong closedness" in set based topological categories are introduced by Baran [2,4] and it is shown in [9] that these notions form an appropriate closure operator in the sense of Dikranjan and Giuli [14] in some well-known topological categories. Moreover, various generalizations of each of T i , i = 0, 1, 2 separation properties for an arbitrary topological category over Set, the category of sets are given and the relationship among various forms of each of these notions are investigated by Baran in [2,7,8,10,11].…”

T3 AND T4-objects in the topological category of cauchy spaces

“…Moreover, various generalizations of each of T i , i = 0; 1; 2 separation properties for an arbitrary topological category over SET, the category of sets are given and the relationship among various forms of each of these notions are investigated by Baran in [2], [7], [8], [10], [12] and [14].…”