Abstract
We define the concepts of 𝐺𝐵-metric in sets over 𝜎-complete Boolean algebra and obtain some applications of them on the theory of topology. We also study some related properties of them.
1. Introduction
Numerous studies have been made concerning geometries and topologies induced in sets by general distance functions. A formulation of the notion generalized metric space [or 𝐺-metric space]" has been given [1]. In this paper, we begin the elaboration of the topology induced in sets over 𝜎-complete Boolean algebra.
In this paper, 𝐵 shall always denote a 𝜎-complete Boolean algebra. In 𝐵, we denote the operations of join, meet, and complement by 𝑎𝑏, 𝑎𝑏 and 𝑎, respectively.
2. The 𝐺𝐵-Metric Spaces
In 1952, a new structure of metric spaces, so called 𝐵-metric space was introduced by Ellis and Sprinkle [2], on the set 𝑋 to Boolean algebra [for details see [3, 4]].
Definition 2.1 [see [2]]. A 𝐵-metric space is a set 𝑋 with a map 𝑑𝑋×𝑋𝐵 [𝐵 is 𝜎-Boolean algebra] with the properties[1]𝑑[𝜉,𝜂]=0 if and only if 𝜉=𝜂,[2]𝑑[𝜂,𝜉]=𝑑[𝜉,𝜂], [symmetry], and [3]𝑑[𝜉,𝜁]𝑑[𝜉,𝜂]𝑑[𝜂,𝜁], for all 𝜉, 𝜂, 𝜁 belong to 𝑋.
In [1], the present author has introduced a new structure of metric spaces which is a generalized idea of the ordinary metric space. The term generalized metric space is used in [5, 6].
Definition 2.2 [see [7]]. Let 𝑋 be a nonempty set and 𝐺𝑋×𝑋×𝑋[0,] be a function satisfying the following properties:[𝐺1]𝐺[𝑥,𝑦,𝑧]=0 if 𝑥=𝑦=𝑧,[𝐺2]0