{
  "id": "1602.06891",
  "version": "v1",
  "published": "2016-02-22T18:59:30.000Z",
  "updated": "2016-02-22T18:59:30.000Z",
  "title": "Suprema of continuous functions on connected spaces",
  "authors": [
    "André Santoleri Villa Barbeiro",
    "Rogério Augusto dos Santos Fajardo"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $K$ be a compact Hausdorff space and let $(f_n)_{n\\in \\N}$ be a pairwise disjoint sequence of continuous functions from $K$ into $[0,1]$. We say that a compact space $L$ \\emph{adds supremum} of $(f_n)_{n\\in \\N}$ in $K$ if there exists a continuous surjection $\\pi:L\\longrightarrow K$ such that there exists $sup\\{f_n\\circ\\pi:n\\in \\N\\}$ in $C(L)$. Moreover, we expect that $L$ preserves suprema of disjoint continuous functions which already existed in $C(K)$. Namely, if $sup\\{g_n:n\\in \\N\\}$ exists in $C(K)$, we must have $sup\\{g_n\\circ\\pi:n\\in \\N\\}$ in $C(L)$. This paper studies the preservation of connectedness in extensions by continuous functions -- a technique developed by Piotr Koszmider to add suprema of continuous functions on Hausdorff connected compact spaces -- proving the following results: (1) If $K$ is a metrizable and locally connected compactum, then any extension of $K$ by continuous functions is connected (but it may be not locally connected). (2) There exists a disconnected extension of a metrizable connected compactum $K$. (3) For any metrizable compactum $K$ there exists a disconnected $L$ which is obtained from $K$ by finitely many extensions by continuous functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-22T18:59:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D05"
    ],
    "keywords": [
      "connected spaces",
      "compact hausdorff space",
      "connected compactum",
      "hausdorff connected compact spaces",
      "add suprema"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160206891S"
    }
  }
}