# dense in-itself

A subset $A$ of a topological space^{} is said to be *dense-in-itself* if $A$ contains no isolated points^{}.

Note that if the subset $A$ is also a closed set^{}, then $A$ will be a perfect set^{}. Conversely, every perfect set is dense-in-itself.

A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers. This set is dense-in-itself because every neighborhood^{} of an irrational number $x$ contains at least one other irrational number $y\ne x$. On the other hand, this set of irrationals is not closed because every rational number lies in its closure^{}.

For similar reasons, the set of rational numbers is also dense-in-itself but not closed.

Title | dense in-itself |
---|---|

Canonical name | DenseInitself |

Date of creation | 2013-03-22 14:38:29 |

Last modified on | 2013-03-22 14:38:29 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 4 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54A99 |

Related topic | ScatteredSpace |