Implications

We have the following indirect implication of form equivalence classes:

385 \(\Rightarrow\) 358
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
385 \(\Rightarrow\) 386 Products, the Baire category theorem, and the axiom of dependent choice, Herrlich-Keremedis-1999a[1999a], Topology and its Applications.
386 \(\Rightarrow\) 10 Products, the Baire category theorem, and the axiom of dependent choice, Herrlich-Keremedis-1999a[1999a], Topology and its Applications.
10 \(\Rightarrow\) 358 clear

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

Howard-Rubin Number Statement
385:

Countable Ultrafilter Theorem:  Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.

386:

Every B compact (pseudo)metric space is Baire.

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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