Implications

We have the following indirect implication of form equivalence classes:

99 \(\Rightarrow\) 280
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
99 \(\Rightarrow\) 70 Handbook of Analysis and its Applications, Schechter, [1996a]
70 \(\Rightarrow\) 142 The Axiom of Choice, Jech, 1973b, page 7 problem 11
142 \(\Rightarrow\) 280 clear

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

Howard-Rubin Number Statement
99:

Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family  of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\).  Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

142:

\(\neg  PB\):  There is a set of reals without the property of Baire.  Jech [1973b], p. 7.

280:

There is a complete separable metric space with a subset which does not have the Baire property.

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