Cop and robber on 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.