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Notes et références
3．Pour d'autres énoncés équivalents à ACω, voir (en) Horst Herrlich, « Choice principles in elementary topology and analysis », Comment. Math. Univ. Carolinae, vol. 38, no 3,‎ 1997, p. 545-552 (lire en ligne [archive]) et (en) Paul Howard et Jean E. Rubin, Consequences of the Axiom of Choice, Providence, R.I., AMS, 1998.

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Comment.Math.Univ.Carolin. 38,3(1997)545–552 545
Choice principles in elementary topology and analysis Horst Herrlich
1. In the realm of the reals
We start by observing that several familiar topological properties of the reals are equivalent to each other and to rather natural choice-principles.
Theorem 1.1 ([15], [29], [30]). Equivalent are:
1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x,
2. a function f : R → R is continuous at a point x iff it is sequentially continuous at x,
3. a real-valued function f : A → R from a subspace A of R is continuous iff it is sequentially continuous,
4. each subspace of R is separable,
5. R is a Lindel¨ of space,
6. Q is a Lindel¨ of space,
7. N is a Lindel¨ of space,
8. each unbounded subset of R contains an unbounded sequence,
9. the Axiom of Choice for countable collections of subsets of R.
There exist models of ZF that violate the above conditions ([17], [18]).
Observe the fine distinction between conditions 2 and 3 of Theorem 1.1.
These may lead one to assume that also the following property is equivalent to the above conditions:
(*) a function f : R −→ R is continuous iff it is sequentially continuous.
However, this would be a serious mistake: (*) holds in ZF (without any choiceassumptions) — see [29].
If, however, we consider functions f : X −→ R with metric domain we need even more choice than in Theorem 1.1, — see Theorem 2.1.
Proposition 1.2 ([15]). Equivalent are:
1. in R, every bounded infinite set contains a convergent injective sequence,
2. every infinite subset of R is Dedekind-infinite.
There exist models of ZF that violate the above conditions ([18]).
Obviously, the conditions of Theorem 1.1 imply the conditions of Proposition 1.2.
Is the converse true?
Observe that the following slight modifications of condition 1 in Proposition 1.2 hold in ZF:
(a) in R, every bounded countable set contains a convergent injective sequence,
(b) in R, for every bounded infinite set there exists an accumulation point.

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