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Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ⊆ B has a complementary subspace S” and “for every family 𝒜 of disjoint odd sized sets there exists a subfamily ℱ={Fj:j ϵω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ℚ every generating set includes a basis”.