# no countable dense subset of a complete metric space is a ${G}_{\delta}$

Let $(X,d)$ be a complete metric space with no isolated points^{}, and let $D\subset X$ be a countable^{} dense set. Then $D$ is not a
${G}_{\delta}$ set (http://planetmath.org/G_deltaSet).

Proof

First, we will prove that $D$ is first category. Then, supposing that $D$ is a ${G}_{\delta}$, we will conclude that $X-D$ must be first category. But then so must be $X$, which is absurd because $X$ is complete^{}.

1) $D$ is a first category set:

By hypothesis^{} $D={\left\{{x}_{i}\right\}}_{i\in N}$. Let’s see that each singleton is nowhere dense if $X$ has no isolated points:

$\overline{\left\{{x}_{i}\right\}}=\left\{{x}_{i}\right\}$ (trivially). Suppose that ${\left\{{x}_{i}\right\}}^{o}=\left\{{x}_{i}\right\}$. Then there is a ball aisolating the point. Absurd ($X$ has no isolated points). Then ${\left\{{x}_{i}\right\}}^{o}=\mathrm{\varnothing}$ and we have that ${\left(\overline{\left\{{x}_{i}\right\}}\right)}^{o}=\mathrm{\varnothing}$ so every singleton is nowhere dense and $D$ is of first category because it is a countable union of nowhere dense sets.

2) Suppose $D$ is a ${G}_{\delta}$, that is, $D=\bigcap _{i=1}^{\mathrm{\infty}}{U}_{i}$ such that every $U$ is open. As $D$ is dense, then each $U$ is dense, because $D\subset {U}_{i}\Rightarrow \overline{D}=X\subset \overline{{U}_{i}}\mathit{\hspace{1em}}\forall i$. But then $X-D=X-\bigcap _{i=1}^{\mathrm{\infty}}{U}_{i}=\bigcup _{i=1}^{\mathrm{\infty}}(X-{U}_{i})$ and ${\left(\overline{X-{U}_{i}}\right)}^{o}={\left(X-{U}_{i}^{o}\right)}^{o}={\left(X-{U}_{i}\right)}^{o}=X-\overline{{U}_{i}}=\mathrm{\varnothing}$

which implies that $X-D$ is of first category. Then $D\bigcup (X-D)=X$ is of first category. Absurd, because $X$ is complete.

Title | no countable dense subset of a complete metric space is a ${G}_{\delta}$ |
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Canonical name | NoCountableDenseSubsetOfACompleteMetricSpaceIsAGdelta |

Date of creation | 2013-03-22 14:59:06 |

Last modified on | 2013-03-22 14:59:06 |

Owner | gumau (3545) |

Last modified by | gumau (3545) |

Numerical id | 8 |

Author | gumau (3545) |

Entry type | Result |

Classification | msc 54E52 |