Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Furtherness in finite topological spaces

Akhilesh Badra, Hemant Kumar Singh

Abstract

In this paper, we introduce a novel distance-like notion of furtherness for finite topological spaces, demonstrating that every finite space can be viewed as an asymmetric pseudometric space. In particular, we show that every finite T0 space is asymmetric metric space. The topology induced by the forward balls coincides with the original topology of the space, while the backward balls induce the opposite topology. To capture essential information about each finite space, we construct a furtherness Matrix, which gives significant structural details of the finite space. As an application, we introduce the notion of center and radius of subsets of finite topological spaces.

Furtherness in finite topological spaces

Abstract

In this paper, we introduce a novel distance-like notion of furtherness for finite topological spaces, demonstrating that every finite space can be viewed as an asymmetric pseudometric space. In particular, we show that every finite T0 space is asymmetric metric space. The topology induced by the forward balls coincides with the original topology of the space, while the backward balls induce the opposite topology. To capture essential information about each finite space, we construct a furtherness Matrix, which gives significant structural details of the finite space. As an application, we introduce the notion of center and radius of subsets of finite topological spaces.
Paper Structure (11 sections, 36 theorems)

This paper contains 11 sections, 36 theorems.

Table of Contents

  1. Introduction
  2. Furtherness function
  3. Preorder on finite space
  4. Furtherness function on product of finite spaces
  5. Topology induced by furtherness
  6. Topology generated by balls
  7. Furtherness matrix
  8. Center and Radius of a subset of finite topologiocal space
  9. Center and Radius of a subset of finite space
  10. Center and radius of union of subsets of a finite space
  11. The Quasi-Center and the Quasi-Radius

Key Result

Theorem 2.6

Let $X$ be a finite $T_0$ space. Then $\Psi(x,y)=\Psi(y,x)=0 \iff x=y,$ for $x,y\in X$.

Theorems & Definitions (74)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Furtherness function
  • Theorem 2.6
  • proof
  • Example 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 64 more