Implications

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 197
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 \(\Rightarrow\) 197 The plane is the union of three rectilinearly accessible sets, Davies, R. O. 1978, Real Anal. Exchange.

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.

197:

\({\Bbb R}^{2}\) is the union of three sets \(C\) with the property that for all \(x\in C\) there is a straight line \(L\) such that \(L\cap C = \{x\}\).

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