Axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that any countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.
The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), (Jech 1973) which in turn is weaker than the axiom of choice (AC). Paul Cohen showed that ACω, is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice (Potter 2004). ACω holds in the Solovay model.
ZF + ACω suffices to prove that the union of countably many countable sets is countable. It also suffices to prove that every infinite set is Dedekind-infinite (equivalently: has a countably infinite subset).
ACω is particularly useful for the development of analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point x of a set S⊆R is the limit of some sequence of elements of S\{x}, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω. For other statements equivalent to ACω, see Herrlich (1997) and Howard & Rubin (1998).
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