Implications

We have the following indirect implication of form equivalence classes:

406 \(\Rightarrow\) 18
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
406 \(\Rightarrow\) 10 The axiom of choice and two particular forms of Tychonoff theorem, Alas, O. T. 1969, Portugal. Math.
10 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 18 clear

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

Howard-Rubin Number Statement
406:

The product of compact Hausdorf spaces is countably compact. Alas [1994].

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.

18:

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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