Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
95-F \(\Rightarrow\) 67	Some theorems on vector spaces and the axiom of choice, Bleicher, M. 1964, Fund. Math. The Axiom of Choice, Jech, 1973b, page 148 problem 10.4
67 \(\Rightarrow\) 147	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26

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

Howard-Rubin Number	Statement
95-F:	Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4.
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).
147:	\(A(D2)\): Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.

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