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Typed topology and its application to data set

Wanjun Hu

TL;DR

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.

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.

Typed topology and its application to data set

TL;DR

Left-r and up-left-r type open sets are introduced for data sets in , so that tracks, port, branches can be calculated.

Abstract

The concept of 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 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, and type open sets are introduced for data sets in , so that tracks, port, branches can be calculated.
Paper Structure (5 sections, 20 theorems, 1 equation, 4 figures, 1 table)

This paper contains 5 sections, 20 theorems, 1 equation, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Typed topological spaces
  3. Clusters as closure
  4. Local Horizontal Axis And Extension
  5. Branches

Key Result

Proposition 2.8

When $\sigma$ has least $p$-neighborhood for every point in $X$, a point $x$ is a $p$-accumulation point of a subset $A$ if and only if $(p\vdash U_{min}(x))\cap A\neq\emptyset$. $\Box$

Figures (4)

  • Figure 1: A cluster that is not straight and locations for surgery
  • Figure 2: Combined indexing on uniformly typed spaces
  • Figure 3: (p,q)-straight examples
  • Figure 4: 2D data set with non symmetrically typed topology

Theorems & Definitions (71)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • proof
  • ...and 61 more