Furtherness in finite topological spaces

TL;DR

This paper introduces a new concept called furtherness to analyze finite topological spaces, showing they can be represented as asymmetric pseudometric spaces and providing tools like the furtherness matrix for structural insights.

Contribution

It proposes the furtherness notion and the furtherness matrix, offering a novel way to represent and analyze finite topological spaces as asymmetric metric spaces.

Findings

Finite spaces can be viewed as asymmetric pseudometric spaces.

Every finite T0 space is an asymmetric metric space.

The furtherness matrix captures key structural information.

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.