Implications

We have the following indirect implication of form equivalence classes:

181 \(\Rightarrow\) 74
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
181 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 74 note-10

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

Howard-Rubin Number Statement
181:

\(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function.

8:

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

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

74:

For every \(A\subseteq\Bbb R\) the following are equivalent:

  1. \(A\) is closed and bounded.
  2. Every sequence \(\{x_{n}\}\subseteq A\) has a convergent subsequence with limit in A.
Jech [1973b], p 21.

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