{
  "id": "2004.02013",
  "version": "v1",
  "published": "2020-04-04T20:43:57.000Z",
  "updated": "2020-04-04T20:43:57.000Z",
  "title": "Games and hereditary Baireness in hyperspaces and spaces of probability measures",
  "authors": [
    "Mikołaj Krupski"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter $\\mathcal{F}$ and prove that it is equivalent to $K(\\mathcal{F})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_r(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$ which is not completely metrizable with $P_r(X)$ hereditarily Baire. As far as we know this is the first example of this kind.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-04T20:43:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "54E52",
      "60B05",
      "91A44",
      "54D40",
      "54D20",
      "54D80"
    ],
    "keywords": [
      "probability measures",
      "hereditarily baire",
      "hereditary baireness",
      "separable metrizable space",
      "hyperspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}