Implications

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 169
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
147 \(\Rightarrow\) 91 The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic
note-26
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 272 Models of set theory containing many perfect sets, Truss, J. K. 1974b, Ann. Math. Logic
272 \(\Rightarrow\) 169 clear

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

Howard-Rubin Number Statement
147:

\(A(D2)\):  Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

272:

There is an \(X\subseteq{\Bbb R}\) such that neither \(X\) nor \(\Bbb R - X\) has a perfect subset.

169:

There is an uncountable subset of \({\Bbb R}\) without a perfect subset.

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