Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
345 \(\Rightarrow\) 14 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
14 \(\Rightarrow\) 153 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
153 \(\Rightarrow\) 10 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
10 \(\Rightarrow\) 358 clear

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

Howard-Rubin Number Statement
345:

Rasiowa-Sikorski Axiom:  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\).

14:

BPI: Every Boolean algebra has a prime ideal.

153:

The closed unit ball of a Hilbert space is compact in the weak topology.

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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