Equivalents of AC

Set Theory · Axiom Academy

LESSON Equivalents of the Axiom of Choice Surprising statements that are all equivalent to AC Statement: If every chain in a partially ordered set has an upper bound, then the poset contains at least one maximal element. Chain: A totally ordered subset Upper bound: An element ≥ all elements in the chain Maximal element: An element with no element strictly greater than it Zorn's Lemma is the most commonly used form of AC in algebra and analysis! Statement: Every set can be well-ordered. That is, for any set S, there exists a total ordering ≤ such that every non-empty subset has a least element. This means even the real numbers ℝ can be well-ordered, though we cannot explicitly construct such an ordering! 3. Hausdorff Maximal Principle Statement: Every partially ordered set contains a maximal chain (a chain that is not properly contained in any other chain). This was historically one of the first equivalents discovered, formulated by Felix Hausdorff in 1914. Statement: The product of any collection of compact topological spaces is compact (in the product topology). This is a fundamental theorem in topology. Remarkably, it's equivalent to the full Axiom of Choice—you cannot prove Tychonoff without AC! These statements all turn out to be equivalent in ZF set theory: The proofs of these equivalences form a beautiful circle: AC ⇒ Zorn ⇒ Hausdorff ⇒ Well-Ordering ⇒ AC.

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