Scattered products in fundamental groupoids

TL;DR

This paper establishes a connection between the well-definedness of scattered infinite products in fundamental groupoids and the homotopically Hausdorff property of spaces, using closure operators on the $$-subgroup lattice.

Contribution

It provides a new characterization of the homotopically Hausdorff property via the well-definedness of scattered products in fundamental groupoids.

Findings

Scattered products in fundamental groupoids are well-defined if and only if the space is homotopically Hausdorff.

Introduces a novel use of closure operators on the $$-subgroup lattice to analyze fundamental groupoids.

Establishes a link between infinitary operations and topological properties of spaces.

Abstract

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products indexed by a scattered linear order in the fundamental groupoid of a first countable space is equivalent to the homotopically Hausdorff property. To prove this characterization, we employ the machinery of closure operators, on the $\pi_1$-subgroup lattice, defined in terms of test maps from one-dimensional domains.