# dense total order

A total order^{} $$ is *dense* if whenever $$ in $S$, there exists at least one element $y$ of $S$ such that $$. That is, each nontrivial closed interval^{} has nonempty interior.

A subset $T$ of a total order $S$ is *dense in* $S$ if for every $x,z\in S$ such that $$, there exists some $y\in T$ such that $$. Because of this, a dense total order $S$ is sometimes said to be *dense in itself*.

For example, the integers with the usual order are not dense, since there is no integer strictly between $0$ and $1$. On the other hand, the rationals $\mathbb{Q}$ are dense, since whenever $r$ and $s$ are rational numbers, it follows that $(r+s)/2$ is a rational number strictly between $r$ and $s$. Also, both $\mathbb{Q}$ and the irrationals $\mathbb{R}\setminus \mathbb{Q}$ are dense in $\mathbb{R}$.

It is usually convenient to assume that a dense order has at least two elements. This allows one to avoid the trivial cases of the one-point order and the empty order.

Title | dense total order |
---|---|

Canonical name | DenseTotalOrder |

Date of creation | 2013-03-22 16:40:48 |

Last modified on | 2013-03-22 16:40:48 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 8 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 06A05 |

Synonym | dense linear order |

Related topic | LinearContinuum |

Defines | dense |

Defines | dense in |

Defines | dense in itself |