Nearness in Digital Images and Proximity Spaces
The concept of “nearness”, which has been dealt with as soon as one started studying digital images, finds one of its rigorous forms in the notion of proximity space. It is this notion, together with “nearness preserving mappings”, that we investigate in this paper. We first review basic examples as they naturally occur in digital topologies, making also brief comparison studies with other concepts in digital geometry. After this we characterize proximally continuous mappings in metric spaces. Finally, we show by example that the “proximite complexity” of a finite covering in a digital picture may be too high to be adequately depicted in a finite topological space. This combinatorial result may indicate another conceptual advantage of proximities over topologies.
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