# examples of nowhere dense sets

Note that $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$ under the usual topology: $\mathrm{int}\overline{\mathbb{Z}}=\mathrm{int}\mathbb{Z}=\mathrm{\varnothing}$. Similarly, $\frac{1}{n}\mathbb{Z}$ is nowhere dense for every $n\in \mathbb{Z}$ with $n>0$.

This result provides an alternative way to prove that $\mathbb{Q}$ is meager in $\mathbb{R}$ under the usual topology, since $\mathbb{Q}={\displaystyle \bigcup _{n\in \mathbb{Z}\text{and}n0}}\frac{1}{n}\mathbb{Z}$.

Title | examples of nowhere dense sets |
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Canonical name | ExamplesOfNowhereDenseSets |

Date of creation | 2013-03-22 17:07:05 |

Last modified on | 2013-03-22 17:07:05 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 4 |

Author | Wkbj79 (1863) |

Entry type | Example |

Classification | msc 54A99 |

Related topic | ExampleOfAMeagerSet |