Axiom Of Choice

We have the following indirect implication of form equivalence classes:

352 \(\Rightarrow\) 358 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 \(\Rightarrow\) 32	L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
32 \(\Rightarrow\) 10	clear
10 \(\Rightarrow\) 358	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.
32:	\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
358:	\(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

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