Implications

We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 404
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 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 404 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:

\(C(\aleph_{0},\infty)\):

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

404:

Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements. Ash [1983].

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