Axiom Of Choice

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 113 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
44 \(\Rightarrow\) 43	The interdependence of certain consequences of the axiom of choice, Levy, A. 1964, Fund. Math. The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1
43 \(\Rightarrow\) 113	note-7

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

Howard-Rubin Number	Statement
44:	\(DC(\aleph _{1})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(\|Y\| < \aleph_{1}\) there is an \(x \in X\) with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).
43:	\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136.
113:	Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.

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