{
  "id": "2102.04856",
  "version": "v1",
  "published": "2021-02-09T14:51:04.000Z",
  "updated": "2021-02-09T14:51:04.000Z",
  "title": "On Axiomatic Characterization of Alexander-Spanier Normal Homology Theory of General Topological Spaces",
  "authors": [
    "Vladimer Baladze",
    "Anzor Beridze",
    "Leonard Mdzinarishvili"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "The Alexandroff-\\v{C}ech normal cohomology theory [Mor$_1$], [Bar], [Ba$_1$],[Ba$_2$] is the unique continuous extension \\cite{Wat} of the additive cohomology theory [Mil], [Ber-Mdz$_1$] from the category of polyhedral pairs $\\mathcal{K}^2_{Pol}$ to the category of closed normally embedded, the so called, $P$-pairs of general topological spaces $\\mathcal{K}^2_{Top}$. In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-\\v{C}ech normal cohomology. Using this fact and methods developed in [Ber-Mdz$_3$] we construct an exact, the so called, Alexander-Spanier normal homology theory on the category $\\mathcal{K}^2_{Top},$ which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs $\\mathcal{K}^2_{C}.$ Moreover, we give an axiomatic characterization of the constructed homology theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-09T14:51:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N07",
      "55N05"
    ],
    "keywords": [
      "alexander-spanier normal homology theory",
      "general topological spaces",
      "axiomatic characterization",
      "alexander-spanier normal cohomology theory",
      "steenrod homology theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}