# compact Hausdorff space is extremally disconnected if its function algebra is a bounded complete lattice

Let $X$ be a compact^{} Hausdorff space and $C(X)$ the algebra of continuous functions^{} $X\u27f6\u2102$. Recall that $C(X)$ is a vector lattice with the usual order (http://planetmath.org/PartialOrder): $f\le g\u27fag-f$ takes positive^{} (or zero) values.

$$

Theorem - If every subset of $C(X)$ that is bounded from above has a least upper bound (i.e. $C(X)$ is a bounded complete lattice^{}), then $X$ is extremally disconnected.

Title | compact Hausdorff space is extremally disconnected if its function algebra is a bounded complete lattice |
---|---|

Canonical name | CompactHausdorffSpaceIsExtremallyDisconnectedIfItsFunctionAlgebraIsABoundedCompleteLattice |

Date of creation | 2013-03-22 17:53:08 |

Last modified on | 2013-03-22 17:53:08 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 54G05 |

Classification | msc 46J10 |

Classification | msc 06F20 |

Synonym | sufficient condition for a compact Hausdorf space to be extremally disconnected |