Axiom Of Choice

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 313 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
147 \(\Rightarrow\) 91	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26
91 \(\Rightarrow\) 313	Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7

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

Howard-Rubin Number	Statement
147:	\(A(D2)\): Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.
91:	\(PW\): The power set of a well ordered set can be well ordered.
313:	\(\Bbb Z\) (the set of integers under addition) is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

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