Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
424 \(\Rightarrow\) 94 On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
94 \(\Rightarrow\) 74 note-10

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

Howard-Rubin Number Statement
424:  Every Lindel\"{o}f metric space is super second countable.  \ac{Gutierres} \cite{2004} and note 159. \iput{super second countable}
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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