Hypothesis: HR 317:
Weak Sikorski Theorem: If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\).
Conclusion: HR 304:
There does not exist a \(T_2\) topological space \(X\) such that every infinite subset of \(X\) contains an infinite compact subset.
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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