Typed topology and its application to data set

TL;DR

This paper introduces the concept of typed topology, applying it to finite data sets in R^2, and demonstrates its usefulness in representing clustering results and defining new topological concepts for data analysis.

Contribution

It presents the novel concept of typed topology and applies it to finite data sets, enabling new ways to analyze clustering and data structure.

Findings

Typed topology effectively models data clustering.

New concepts like tracks and branches are introduced for data analysis.

Typed topology provides a framework for analyzing data in R^2.

Abstract

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 $R^2$ is a typically typed topological space. Clusters calculated by the DBSCAN algorithm for data clustering can be well represent 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 a finite typed topological space. Finally, $left-r$ and $up-left-r$ type open sets are introduced for data sets in $R^2$, so that tracks, port, branches can be calculated.