Typed topology and its application to data sets

The concept of typed topology is introduced. In a typed topological space, some open sets are assigned types, and topological concepts such as closure, connectedness can be defined using types. A finite data set in R2 is a typically typed topological space. Clusters calculated by the DBSCAN algorithm for data clustering can be well represented in a finite typed topological space. Other concepts such as tracks, port (starting points), type-p-connectedness, p-closure-connectedness, indexing, branches are also introduced for finite typed topological spaces. Finally, left - r and up - left - r type open sets are introduced for finite data sets in R2, so that tracks, port, branches can be calculated.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

The concept of typed topology is introduced, and its application in finite data sets and data clustering is discussed. Several related concepts, such as tracks, ports, and branches, are introduced for finite typed topological spaces.