Scattered products in fundamental groupoids

Brazas, Jeremy

doi:10.1090/proc/14883

Scattered products in fundamental groupoids
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Proc. Amer. Math. Soc. 148 (2020), 2655-2670 Request permission

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.

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