Implications

We have the following indirect implication of form equivalence classes:

352 \(\Rightarrow\) 182
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
352 \(\Rightarrow\) 31 On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
31 \(\Rightarrow\) 34 clear
34 \(\Rightarrow\) 104 clear
104 \(\Rightarrow\) 182 clear

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

Howard-Rubin Number Statement
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.

34:

\(\aleph_{1}\) is regular.

104:

There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

182:

There is an aleph whose cofinality is greater than \(\aleph_{0}\).

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