# proof of fixed points of normal functions

Suppose $f$ is a $\kappa $-normal function and consider any $$ and define a sequence by ${\alpha}_{0}=\alpha $ and ${\alpha}_{n+1}=f({\alpha}_{n})$. Let $$. Then, since $f$ is continuous,

$$ |

So $\mathrm{Fix}(f)$ is unbounded^{}.

Suppose $N$ is a set of fixed points^{} of $f$ with $$. Then

$$f(supN)=\underset{\alpha \in N}{sup}f(\alpha )=\underset{\alpha \in N}{sup}\alpha =supN$$ |

so $supN$ is also a fixed point of $f$, and therefore $\mathrm{Fix}(f)$ is closed.

Title | proof of fixed points of normal functions |
---|---|

Canonical name | ProofOfFixedPointsOfNormalFunctions |

Date of creation | 2013-03-22 13:29:01 |

Last modified on | 2013-03-22 13:29:01 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 4 |

Author | Henry (455) |

Entry type | Proof |

Classification | msc 03E10 |