Implications

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 212
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 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 212 clear

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.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

212:

\(C(2^{\aleph_{0}},\subseteq{\Bbb R})\): If \(R\) is a relation on \({\Bbb R}\) such that for all \(x\in{\Bbb R}\), there is a \(y\in{\Bbb R}\) such that \(x\mathrel R y\), then there is a function \(f: {\Bbb R} \rightarrow{\Bbb R}\) such that for all \(x\in{\Bbb R}\), \(x\mathrel R f(x)\).

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