Hypothesis: HR 0: \(0 = 0\).
Conclusion: HR 78:
Urysohn's Lemma: If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N3\) Mostowski's Linearly Ordered Model | \(A\) is countably infinite;\(\precsim\) is a dense linear ordering on \(A\) without first or lastelements (\((A,\precsim) \cong (\Bbb Q,\le)\)); \(\cal G\) is the group of allorder automorphisms on \((A,\precsim)\); and \(S\) is the set of all finitesubsets of \(A\) |
| \(\cal N8\) L\"auchli's Model II | \(A\) is countably infinite, ordered likethe rational numbers; \(\cal G\) is the group of all order automorphisms of\(A\); and \(S\) is the set of all subsets \(E\) of \(A\) such that \(E\) has atmost a finite number of accumulation points and every infinite subset of\(E\) has an accumulation point |
Code: 3
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