Implications

We have the following indirect implication of form equivalence classes:

343 \(\Rightarrow\) 73
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
343 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 11 clear
11 \(\Rightarrow\) 12 clear
12 \(\Rightarrow\) 73 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
Set Theory: Techniques and Applications, De la Cruz/Di Prisco, 1998b, 47-70

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

Howard-Rubin Number Statement
343:

A product of non-empty, compact \(T_2\) topological spaces is non-empty.

62:

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

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

11:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

12:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

73:

\(\forall n\in\omega\), \(PC(\infty,n,\infty)\):  For every \(n\in\omega\), if \(C\) is an infinite family of \(n\) element sets, then \(C\) has an infinite subfamily with a choice function. De la Cruz/Di Prisco [1998b]

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