Axiomatic Approach

Abstract

Starting with an intuitive concept of “nearness” as a binary relation, semi-proximity spaces (sp-spaces) are defined. The restrictions on semi-proximity spaces are weaker than the restrictions on topological proximity spaces. Thus, semi-proximity spaces generalize classical topological spaces. We will use semi-proximity spaces to establish a formal relationship between the topological concepts of digital image processing and their continuous counterparts in ℝⁿ. This is possible, since ℝⁿ with the usual topology and digital images with their usual structure are sp-spaces. Examples of different semi-proximity relations on digital images are given which induce the usual connectedness on digital images. This is not possible in classical topology.