# normal (ordinal) function

###### Definition.

A function^{} $F:\mathrm{\mathbf{O}\mathbf{n}}\to \mathrm{\mathbf{O}\mathbf{n}}$ is *continuous ^{}* if and only if for each $u\subset \mathrm{\mathbf{O}\mathbf{n}}$ such that $u\ne \mathrm{\varnothing}$ it holds that $F(sup(u))=sup\{F(\alpha )|\alpha \in u\}$.

###### Definition.

A function $F:\mathrm{\mathbf{O}\mathbf{n}}\to \mathrm{\mathbf{O}\mathbf{n}}$ is *order preserving* if and only if for each $\alpha ,\beta \in \mathrm{\mathbf{O}\mathbf{n}}$ such that $$ it follows that $$.

###### Definition.

A function $F:\mathrm{\mathbf{O}\mathbf{n}}\to \mathrm{\mathbf{O}\mathbf{n}}$ is a *normal* function if and only if $F$ is continuous and order preserving.

Title | normal (ordinal) function |
---|---|

Canonical name | NormalordinalFunction |

Date of creation | 2013-03-22 15:33:10 |

Last modified on | 2013-03-22 15:33:10 |

Owner | florisje (7763) |

Last modified by | florisje (7763) |

Numerical id | 7 |

Author | florisje (7763) |

Entry type | Definition |

Classification | msc 03E10 |

Defines | continuous (for ordinal functions) |

Defines | order preserving (for ordinal functions) |

Defines | normality |

Defines | normal function |