# $\mathrm{cf}(\mathrm{cf}\alpha )=\mathrm{cf}\alpha $

Let

$\mathrm{cf}\alpha =\beta $ and $$ be cofinal in $\alpha $, and

$\mathrm{cf}\beta =\gamma $ and $$ be cofinal in $\beta $.

The claim of the theorem $\mathrm{cf}(\mathrm{cf}\alpha )=\mathrm{cf}\alpha $ means that $\gamma =\beta $; we prove this fact.

Suppose $\gamma \ne \beta $. Then $$ by $\mathrm{cf}\delta \le \delta $.

Now, $$ is seen to be confinal in $\alpha $, which means that $$, a contradiction^{}. Therefore, $\gamma =\beta $.

Title | $\mathrm{cf}(\mathrm{cf}\alpha )=\mathrm{cf}\alpha $ |
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Canonical name | operatornamecfoperatornamecfalphaoperatornamecfalpha |

Date of creation | 2013-03-22 18:11:22 |

Last modified on | 2013-03-22 18:11:22 |

Owner | yesitis (13730) |

Last modified by | yesitis (13730) |

Numerical id | 8 |

Author | yesitis (13730) |

Entry type | Proof |

Classification | msc 03E04 |