Axiom of countable choice

The axiom stating that every countable family of nonempty sets has a choice function selecting one element from each set.

Choice function: A function that selects one element from each set in a specified family. The axiom asserts that such a function exists for every countable family of nonempty sets.
Countable union of countable sets: The theorem that a countable union of countable sets is countable. Countable choice suffices to select enumerations needed to combine countably many countable sets.
Countable choice and real analysis: The study of which real-analysis theorems require countable choice or weaker selection principles. It tracks the axiom's role in proofs involving sequences, completeness, and countable constructions.
Countable dependent choice: The principle that dependent choices can be made along every sequence of nonempty successor sets. It formalizes countable recursive selection and is closely related to countable choice.
Countable family of sets: A family of sets indexed by a finite or countably infinite set. Countability is the restriction that distinguishes this axiom from unrestricted choice.
Sequential compactness: A property requiring every sequence in a space to have a convergent subsequence. Some proofs extracting a subsequence from infinitely many successive choices invoke countable choice.
Countable choice and countable unions: The relationship between countable choice and cardinality results about countable unions. Without countable choice, a countable union of countable sets need not be countable in every model of set theory.
Boolean prime ideal theorem: The principle that every nontrivial Boolean algebra has a prime ideal. It is weaker than full choice, offering a useful comparison among set-theoretic choice principles.
Axiom of dependent choice: The principle that permits recursively choosing elements from nonempty sets, with each choice allowed to depend on the previous one. It supports dependent recursive selections and is stronger than countable choice in standard set theory.
Baire category theorem: A theorem asserting that complete metric spaces are not countable unions of nowhere-dense sets. Standard proofs can use countable choice to select points from successive dense open sets.

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