In mathematics, a closed preordered set is one whose anti-well-ordered subsets have lower bounds.
Definition Let be a cardinal. A preordered set is called -closed if every subset of whose opposite is well-ordered with order-type less than has a lower bound.
A preordered set is -closed if it is -closed for every . A preordered set is called closed or -closed if it is -closed for every .
A preordered set is inductive if every chain has an upper bound. Since every totally ordered set has a well-ordered cofinal subset, this is equivalent to saying that the preordered set is the opposite of a closed preordered set.
Properties Inductive preordered sets satisfy Zorn's lemma and the Bourbaki–Witt theorem.
A -closed forcing preserves cofinalities less than or equal to , hence cardinals less than or equal to .
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