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 39:
\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N1\) The Basic Fraenkel Model | The set of atoms, \(A\) is denumerable; \(\cal G\) is the group of all permutations on \(A\); and \(S\) isthe set of all finite subsets of \(A\) |
Code: 5
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