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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].
MSC:
 03E25 Axiom of choice and related propositions 54C35 Function spaces in general topology 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 03E60 Determinacy principles