Axiom Of Choice

We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 182 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
426 \(\Rightarrow\) 8	On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
8 \(\Rightarrow\) 27	clear
27 \(\Rightarrow\) 31	clear
31 \(\Rightarrow\) 34	clear
34 \(\Rightarrow\) 104	clear
104 \(\Rightarrow\) 182	clear

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

Howard-Rubin Number	Statement
426:	If \((X,\cal T) \) is a first countable topological space and \((\cal B(x))_{x\in X}\) is a family such that for all \(x \in X\), \(\cal B(x)\) is a local base at \(x\), then there is a family \(( \cal V(x))_{x\in X}\) such that for every \(x \in X\), \(\cal V(x)\) is a countable local base at \(x\) and \(\cal V(x) \subseteq \cal B(x)\).
8:	\(C(\aleph_{0},\infty)\):
27:	\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.
31:	\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.
34:	\(\aleph_{1}\) is regular.
104:	There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.
182:	There is an aleph whose cofinality is greater than \(\aleph_{0}\).

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