{
  "id": "1909.06906",
  "version": "v1",
  "published": "2019-09-15T23:22:29.000Z",
  "updated": "2019-09-15T23:22:29.000Z",
  "title": "Scattered products in fundamental groupoids",
  "authors": [
    "Jeremy Brazas"
  ],
  "categories": [
    "math.AT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-15T23:22:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M05",
      "08A65",
      "55Q52",
      "20L05",
      "55Q05"
    ],
    "keywords": [
      "fundamental groupoid",
      "scattered products",
      "countably infinite linear orders",
      "first countable space",
      "closure operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}