{
  "id": "1702.00326",
  "version": "v1",
  "published": "2017-02-01T15:57:39.000Z",
  "updated": "2017-02-01T15:57:39.000Z",
  "title": "Resolvability in c.c.c. generic extensions",
  "authors": [
    "Lajos Soukup",
    "Adrienne Stanley"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GN",
    "math.LO"
  ],
  "abstract": "Every crowded space $X$ is ${\\omega}$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ of the ground model. We investigate what we can say about ${\\lambda}$-resolvability in c.c.c-generic extensions for ${\\lambda}>{\\omega}$? A topological space is \"monotonically $\\omega_1$-resolvable\" if there is a function $f:X\\to {\\omega_1}$ such that $$\\{x\\in X: f(x)\\ge {\\alpha} \\}\\subset^{dense}X $$ for each ${\\alpha}<{\\omega_1}$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is ${\\omega}_1$-resolvable in some c.c.c-generic extension, (2) $X$ is monotonically $\\omega_1$-resolvable. (3) $X$ is ${\\omega}_1$-resolvable in the Cohen-generic extension $V^{Fn({\\omega_1},2)}$. We investigate which spaces are monotonically $\\omega_1$-resolvable. We show that if a topological space $X$ is c.c.c, and ${\\omega}_1\\le \\Delta(X)\\le |X|<{\\omega}_{\\omega}$, then $X$ is monotonically $\\omega_1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=\\Delta(Y)=\\aleph_\\omega$ which is not monotonically $\\omega_1$-resolvable. The characterization of ${\\omega_1}$-resolvability in c.c.c generic extension raises the following question: is it true that crowded spaces from the ground model are ${\\omega}$-resolvable in $V^{Fn({\\omega},2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is ${\\omega}$-resolvable in $V^{Fn({\\omega},2)}$, (ii) if there is no weakly inaccesssible cardinals, then every crowded space $X$ is ${\\omega}$-resolvable in $V^{Fn({\\omega}_1,2)}$. On the other hand, it is also consistent that there is a crowded space $X$ with $|X|=\\Delta(X)={\\omega_1}$ such that $X$ remains irresolvable after adding a single Cohen real.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-01T15:57:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A35",
      "03E35",
      "54A25"
    ],
    "keywords": [
      "resolvable",
      "crowded space",
      "resolvability",
      "ground model",
      "c-generic extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}