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 157:
Theorem of Goodner: A compact \(T_{2}\) space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of \(C(X)\) (set of continuous real valued functions on \(X\)) which is pointwise bounded has a supremum.
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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