Hypothesis: HR 381:
DUM: The disjoint union of metrizable spaces is metrizable.
Conclusion: HR 156:
Theorem of Gelfand and Kolmogoroff: Two compact \(T_2\) spaces are homeomorphic if their rings of real valued continuous functions are isomorphic.
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 \(A\) |
Code: 5
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