Hypothesis: HR 135:
If \(X\) is a \(T_2\) space with at least two points and \(X^{Y}\) is hereditarily metacompact then \(Y\) is countable. (A space is metacompact if every open cover has an open point finite refinement. If \(B\) and \(B'\) are covers of a space \(X\), then \(B'\) is a refinement of \(B\) if \((\forall x\in B')(\exists y\in B)(x\subseteq y)\). \(B\) is point finite if \((\forall t\in X)\) there are only finitely many \(x\in B\) such that \(t\in x\).) van Douwen [1980]
Conclusion: HR 111:
\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2(\aleph_{\alpha})\) Jech's Model | This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\) |
Code: 5
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