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548: 現代数学の系譜 雑談 ◆yH25M02vWFhP 01/23(木)18:27 ID:OWxAi42s(11/12) AAS
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(参考 追加)
en.wikipedia.org/wiki/Axiom_of_countable_choice
Axiom of countable choice
Equivalent forms
There are many equivalent forms to the axiom of countable choice, in the sense that any one of them can be proven in ZF assuming any other of them. They include the following:[8][9]
・Every countable collection of non-empty sets has a choice function.[8]
・Every infinite collection of non-empty sets has an infinite sub-collection with a choice function.[8]
・Every σ-compact space (the union of countably many compact spaces) is a Lindelöf space (every open cover has a countable subcover).[8] A metric space is σ-compact if and only if it is Lindelöf.[9]
・Every second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset).[8] A metric space is separable if and only if it is σ-compact.[9]
・Every sequentially continuous real-valued function in a metric space is a continuous function.[8]
・Every accumulation point of a subset of a metric space is a limit of a sequence of points from the subset.[9]
・The Rasiowa–Sikorski lemma MA(ℵ0), a countable form of Martin's axiom: in a preorder with the countable chain condition, every countable family of dense subsets has a filter intersecting all the subsets. (In this context, a set is called dense if every element of the preorder has a lower bound in the set.)[8]

References
8^ Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-0977-8. See in particular Form 8, p. 17–18.
9^ Herrlich, Horst (1997). "Choice principles in elementary topology and analysis" (PDF). Comment. Math. Univ. Carolinae. 38 (3): 545. See, in particular, Theorem 2.4, pp. 547–548.

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