The naturals are Lindelöf iff Ascoli holds.

*(English)*Zbl 0983.03039
Koslowski, Jürgen (ed.) et al., Categorical perspectives. Papers from the international conference held in honor of George E. Strecker on the occasion of his 60th birthday at Kent State University, Kent, OH, USA, August 1998. Boston, MA: Birkhäuser. Trends in Mathematics. 191-196 (2001).

The author demonstrates that in ZF set theory the statement (a) “The discrete space of natural numbers is Lindelöf” is equivalent to (b) the classical Ascoli Theorem: “A set \(F\) of continuous selfmaps of the reals is (\(\alpha\)) bounded and equicontinuous iff (\(\beta\)) every sequence in \(F\) has a subsequence that converges continuously to some continuous selfmap of the reals”. Moreover, he shows that in ZF a modified Ascoli Theorem holds which is obtained by replacing the above condition (\(\alpha\)) by the condition (\(\alpha')\) “Each countable subset of \(F\) is equicontinuous and bounded”. In addition, the author presents a weakened form of the axiom of determinateness that is equivalent to the condition (a) above.

For the entire collection see [Zbl 0966.00025].

For the entire collection see [Zbl 0966.00025].

Reviewer: Horst Herrlich (Bremen)

##### MSC:

03E25 | Axiom of choice and related propositions |

54C35 | Function spaces in general topology |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

03E60 | Determinacy principles |

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\textit{Y. T. Rhineghost}, in: Categorical perspectives. Papers from the international conference held in honor of George E. Strecker on the occasion of his 60th birthday at Kent State University, Kent, OH, USA, August 1998. Boston, MA: Birkhäuser. 191--196 (2001; Zbl 0983.03039)