Implications

We have the following indirect implication of form equivalence classes:

28-p \(\Rightarrow\) 375
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
28-p \(\Rightarrow\) 427 clear
427 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 375 note-139

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

Howard-Rubin Number Statement
28-p:

(Where \(p\) is a prime) AL20(\(\mathbb Z_p\)): Every vector space \(V\) over \(\mathbb Z_p\) has the property that every linearly independent subset can be extended to a basis.  (\(\mathbb Z_p\) is the \(p\) element field.) Rubin, H./Rubin, J. [1985], p. 119, Statement AL20

427: \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis.  \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

375:

Tietze-Urysohn Extension Theorem: If \((X,T)\) is a normal topological space, \(A\) is closed in \(X\), and \(f: A\to [0,1]\) is continuous, then there exists a continuous function \(g: X\to [0,1]\) which extends \(f\).

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