Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
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
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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