  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.
