Description:
Form 67 (\(MC\), Multiple Choice Axiom) implies Form 375 (Tietze-Urysohn Extension Theorem).
Content:
We give a proof that Form 67 implies Form 375. To the best of our knowledge, this result is due to David Pincus.
Form 67 (\(MC\), Multiple Choice Axiom) implies Form 375 (Tietze-Urysohn Extension Theorem)
Suppose \(X\) is a normal topological space. Then, if \(A\) and \(B\) are disjoint closed sets, there exist disjoint open sets \(C\) and \(D\) such that \(A\subseteq C\) and \(B\subseteq D\). \(MC\) gives us a rule for choosing a finite number of pairs \((C,D)\) with the above property. Since the intersection of a finite number of open sets is open, \(MC\) implies that there is a function \(F\) such that for each pair ofdisjoint closed sets \((A,B)\), \(F(A,B)=(C,D)\), where \(C\) and \(D\) are disjoint open sets such that \(A\subseteq C\) and \(B\subseteq D\). Thus, using the standard proof of Urysohn's lemma (see, for example, Kelley [1955], p 115), there is a rule for choosing a function \(f: X\to [0,1]\) which is zero on \(A\) and one on \(B\). Given this rule for choosing Urysohn's function, the standard proof of the Tietze-Urysohn Extension Theorem can be given in \(ZF^{0}\) (See, for example, Kelley [1955], p 242.).
Howard-Rubin number:
139
Type:
Theorem
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