{
  "id": "2010.08004",
  "version": "v1",
  "published": "2020-10-15T20:06:27.000Z",
  "updated": "2020-10-15T20:06:27.000Z",
  "title": "$P$-bases and Topological Groups",
  "authors": [
    "Ziqn Feng"
  ],
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "A topological space $X$ is defined to have a neighborhood $P$-base at any $x\\in X$ from some poset $P$ if there exists a neighborhood base $(U_p[x])_{p\\in P}$ at $x$ such that $U_p[x]\\subseteq U_{p'}[x]$ for all $p\\geq p'$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\\mathcal{K}(M)$-base for some separable metric space $M$. This gives a positive answer to Problem 8.6.8 in \\cite{Banakh2019}. Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base. It is shown that any Fr\\'{e}che-Urysohn topological group with a $\\mathcal{K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre~$(\\omega_1, \\omega)$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\\leq \\mathfrak{d}$ such that the precompact subsets are metrizable but $G$ doesn't have an $\\omega^\\omega$-base if $\\omega_1<\\mathfrak{d}$. This gives a consistent negative answer to Problem 6.5 in \\cite{GKL15}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-15T20:06:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free abelian topological group",
      "separable metric space",
      "precompact subset",
      "neighborhood base",
      "consistent negative answer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}