Local Near Sets: Pattern Discovery in Proximity Spaces

The focus of this article is on various approaches to discerning patterns in nonempty sets endowed with a proximity (nearness) relation. Patterns arise in repetitions of some form in the arrangement of the parts of a set. To simplify the steps leading to pattern discovery, an approach inspired by M. Katĕtov is used, where one proximitises certain parts of a nonempty set, rather than proximitise the whole set. In effect, this is a divide-and-conquer approach to pattern discovery. This leads to a study of patterns that are collections of near sets. An important practical outcome of this approach is the discovery of patterns in proximity spaces.