AC implies DC implies countable choice

Depends on

• The Axiom of Choice
• The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
• The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
• The recursion theorem
• Every natural-number-indexed list of nonempty sets has a choice function on its family of values
• Choice for pairs and countable finite choice

Used by

• Conditional flow matching for inverse posteriors Algorithm
• Truncated Milnor invariants from a based link diagram Algorithm
• A degree-zero line bundle with a nonzero section is trivial Corollary
• All four classical braid models realize the Artin presentation Corollary
• Bounded harmonic functions have L-infinity Fatou boundary data Corollary
• Deformation cohomology of a smooth scheme: tangent, obstruction and automorphism spaces Corollary
• Embedding obstructions include all immersion normal-class obstructions Corollary
• Enhanced A-polynomial and Alexander degree identify torus knots Corollary
• Euler number ±1 makes the Milnor sphere bundle a homotopy seven-sphere Corollary
• Finite-dimensional coherent cohomology over a field Corollary
• First Cousin problem on a pseudoconvex domain Corollary
• High normal Pontryagin classes obstruct low-codimension immersions Corollary
• High normal Stiefel-Whitney classes obstruct low-codimension immersions Corollary
• Index-one trajectory moduli spaces are finite Corollary
• Knot Alexander normalization, symmetry, and genus bound Corollary
• Local integrability of measurable conformal structures Corollary
• Minkowski convex-body theorem at equality Corollary
• Nontrivial degree-zero line bundles have no sections Corollary
• Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• The knot Alexander polynomial multiplies under connected sum Corollary
• The Milnor homotopy seven-spheres are homeomorphic to S⁷ Corollary
• The pure braid extension splits as a semidirect product Corollary
• The Solovay model has no Banach–Tarski decomposition Corollary
• The Solovay model has no Vitali or Bernstein set Corollary
• Unordered planar configuration spaces are aspherical Corollary
• Vanishing of the deformation tangent space forces rigidity of deformation classes Corollary
• A decomposable color with nonscalar twist has no single writhe correction Counterexample
• A global section of the quotient that does not lift, and its nonzero connecting class Counterexample
• A homogeneous quotient without invariant measure Counterexample
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• An abstract annulus need not be a smooth concordance Counterexample
• An immersed disk in the four-ball cannot always be cleaned relative to its boundary Counterexample
• Conjugacy alone does not classify braid closures Counterexample
• Degree 2g does not force very ampleness Counterexample
• Degree 2g-1 does not force base-point-freeness Counterexample
• Equal zero-traces do not force smooth concordance Counterexample
• Forgetting framing identifies distinct skein classes Counterexample
• Forgetting the compact-open topology destroys Pontryagin duality Counterexample
• Infinite-dimensional Gaussian random-walk reverse flow is singular Counterexample

…and 449 more results.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Jech, The Axiom of Choice, §2.4, pp.22–23 (standard reference, not scraped)