{
  "id": "1703.08731",
  "version": "v1",
  "published": "2017-03-25T18:55:50.000Z",
  "updated": "2017-03-25T18:55:50.000Z",
  "title": "NIP formulas and Baire 1 definability",
  "authors": [
    "Karim Khanaki"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In this short note, using results of Bourgain, Fremlin, and Talagrand \\cite{BFT}, we show that for a countable structure $M$ and a formula $\\phi(x,y)$ the following are equivalent: (i) $\\phi(x,y)$ has NIP on $M$ (see Definition~\\ref{NIP-formula} below). (ii) For every net $(a_i)_{i\\in I}$ of elements of $M$ and a saturated elementary extension $M^*$ of $M$, if $tp_\\phi(a_i/M^*)\\to p'$, then the type $p'$ is Baire 1 definable over $M$. This implies, as it is well known, that if $\\phi$ is NIP then every global $M$-invariant $\\phi$-type is Baire 1 definable over $M$. Then, we point out that Poizat's result about the numbers of coheirs of types in NIP theories holds in the framework of continuous logic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-25T18:55:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nip formulas",
      "definability",
      "nip theories holds",
      "poizats result",
      "short note"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}