# infimum

The *infimum ^{}* of a set $S$ is the greatest lower bound of $S$ and is denoted $inf(S)$.

Let $A$ be a set with a partial order^{} $\le $, and let $S\subseteq A$. For any $x\in A$, $x$ is a lower bound of $S$ if $x\le y$ for any $y\in S$. The infimum of $S$, denoted $inf(S)$, is the greatest such lower bound; that is, if $b$ is a lower bound of $S$, then $b\le inf(S)$.

Note that it is not necessarily the case that $inf(S)\in S$. Suppose $S=(0,1)$; then $inf(S)=0$, but $0\notin S$.

Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.

Title | infimum |

Canonical name | Infimum |

Date of creation | 2013-03-22 11:48:09 |

Last modified on | 2013-03-22 11:48:09 |

Owner | vampyr (22) |

Last modified by | vampyr (22) |

Numerical id | 11 |

Author | vampyr (22) |

Entry type | Definition |

Classification | msc 06A06 |

Classification | msc 03D20 |

Related topic | Supremum^{} |

Related topic | LebesgueOuterMeasure |

Related topic | MinimalAndMaximalNumber |

Related topic | InfimumAndSupremumForRealNumbers |

Related topic | NondecreasingSequenceWithUpperBound |