Axiom Of Choice

We have the following indirect implication of form equivalence classes:

239 \(\Rightarrow\) 78 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
239 \(\Rightarrow\) 427	clear
427 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 106	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
106 \(\Rightarrow\) 78	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
239:	AL20(\(\mathbb Q\)): Every vector \(V\) space over \(\mathbb Q\) has the property that every linearly independent subset of \(V\) can be extended to a basis. Rubin, H./Rubin, J. [1985], p.119, AL20.
427:	\(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis. \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
106:	Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.
78:	Urysohn's Lemma: If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.

Comment:

Back