Implications

We have the following indirect implication of form equivalence classes:

154 \(\Rightarrow\) 389
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
154 \(\Rightarrow\) 10
10 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 389 clear

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

Howard-Rubin Number Statement
154:

Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

389:

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

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