Implications

We have the following indirect implication of form equivalence classes:

16 \(\Rightarrow\) 209
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
16 \(\Rightarrow\) 352 On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
352 \(\Rightarrow\) 31 On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
31 \(\Rightarrow\) 209 note-72

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

Howard-Rubin Number Statement
16:

\(C(\aleph_{0},\le 2^{\aleph_{0}})\):  Every denumerable collection of non-empty sets  each with power \(\le  2^{\aleph_{0}}\) has a choice function.

352:

A countable product of second countable spaces is second countable.

31:

\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.

209:

There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets.

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