Path spaces I: A Menger-type result

Abstract

Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case of graph-like spaces. Here we introduce a more combinatorial answer, which we call path space, and prove a version of Menger's theorem for it. Since there are many topological path-like objects which induce path spaces, this result can be applied in a variety of settings.