Description:
Form 133 implies Form 340
Content:
A proof that Form 133 implies Form 340. (This proof is due to Eric Hall.) Form 133 is the statement: Every set that cannot be well ordered has an amorphous subset. (This is inf 2 in Rubin/Rubin [1985] where it is shown that inf 2 is equivalent to \(AC\) in \(ZF\), but not in \(ZF^{0}\).) Form 133 is true in \(\cal N1\), \(\cal N24\), \(\cal N24(n)\), and \(\cal N26\). Form 340 is: Every Lindelöf metric space is separable. We shall show that
Form 133 implies Form 340
Suppose \(X\) with the metric \(d\) is a Lindelöf metric space. We shall prove, assuming Form 133, that \(X\) is separable. First we shall show:
1. If \(M\) is an amorphous subset of \(X\), then the range of \(d/M\times M\ (= d[M\times M]\)), is finite.
For each \(m\in M\), \(d[{m}\times M]\) is finite because \(M\) is amorphous. Thus,
\[d[M\times M] =\bigcup_{m\in M}d[{m}\times M]\]
is a finite union of finite sets, so it is finite.
2. If \(X\) has an amorphous subset then \(X\)is not Lindelöf.
Let \(M\) be an amorphous subset of \(X\) and let\(S = {d(a,b): a,b\in M}\). By Lemma 1, \(S\) is a finite set and it is clearly non-empty because \(M\) is infinite. Choose a positive \(\epsilon\) which is less than \(\frac12\) of the minimum number in \(S\). It follows from Lemma 1 that \(\epsilon\)exists. Let \(B(\epsilon,m)\), where \(m\in M\), be an open neighborhood about \(m\) with radius \(\epsilon\). Then
\[{B(\epsilon, m): m\in M}\cup {X \setminus \overline{M}}\]
is an infinite open cover with no countable subcover, Thus, \(X\) is not Lindelöf.
It follows from Lemma 2 and Form 133 that \(X\) can be well ordered.Thus, since \(X\) is a Lindelöf metric space, it follows from form [8 V] that \(X\) is separable.
Howard-Rubin number:
157
Type:
Proof
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