Cop and robber on finite spaces

TL;DR

This paper investigates the conditions under which a cop can guarantee capturing a robber in a topological space without visual contact, focusing on finite spaces and their path properties.

Contribution

It characterizes spaces where a fixed path intersects all other paths, providing new insights into path existence and exotic examples in finite topological spaces.

Findings

Finite spaces with such path properties are characterized.

Existence of universal curves in certain topological spaces.

Discovery of exotic examples of path behaviors in finite spaces.

Abstract

A cop tries to capture a robber in a topological space $X$ being unable to see him. For which spaces $X$ does the cop have a strategy which allows him to capture the robber independently of his efforts to escape? In other words, when is there a curve $\gamma: \mathbb{R}_{\ge 0}\to X$ which has a coincidence with any other curve in $X$. We analyze in particular the case of finite topological spaces and discover general results and exotic examples about paths in these spaces.