# directed set

A *directed set* is a partially ordered set^{} $(A,\le )$ such that whenever $a,b\in A$ there is an $x\in A$ such that $a\le x$ and $b\le x$.

A subset $B\subseteq A$ is said to be *residual* if there is $a\in A$ such that $b\in B$ whenever $a\le b$, and *cofinal* if for each $a\in A$ there is $b\in B$ such that $a\le b$.

A directed set is sometimes called an *upward-directed set*.
We may also define the dual notion:
a *downward-directed set* (or *filtered set*) is a partially ordered set $(A,\le )$ such that whenever $a,b\in A$ there is an $x\in A$ such that $x\le a$ and $x\le b$.

Note: Many authors do not require $\le $ to be antisymmetric,
so that it is only a pre-order (rather than a partial order^{})
with the given property.
Also, it is common to require $A$ to be non-empty.

Title | directed set |

Canonical name | DirectedSet |

Date of creation | 2013-03-22 12:54:00 |

Last modified on | 2013-03-22 12:54:00 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 11 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 06A06 |

Synonym | upward-directed set |

Synonym | upward directed set |

Related topic | Cofinality |

Related topic | AccumulationPointsAndConvergentSubnets |

Defines | residual |

Defines | cofinal |

Defines | downward-directed set |

Defines | downward directed set |

Defines | filtered set |