Axiom Of Choice

We have the following indirect implication of form equivalence classes:

14 \(\Rightarrow\) 146 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
14 \(\Rightarrow\) 49	A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30	clear
30 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 146	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26

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

Howard-Rubin Number	Statement
14:	BPI: Every Boolean algebra has a prime ideal.
49:	Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78.
30:	Ordering Principle: Every set can be linearly ordered.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
146:	\(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is an A1 space. (\((X,T)\) is A1 means if \(U \subseteq T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\)

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