This non-implication,
Form 135 \( \not \Rightarrow \)
Form 146,
whose code is 6,
is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 135 | <p> 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 <em>metacompact</em> if every open cover has an open point finite refinement. If \(B\) and \(B'\) are covers of a space \(X\), then \(B'\) is a <em>refinement</em> of \(B\) if \((\forall x\in B')(\exists y\in B)(x\subseteq y)\). \(B\) is <em>point finite</em> if \((\forall t\in X)\) there are only finitely many \(x\in B\) such that \(t\in x\).) <a href="/excerpts/van-Douwen-1980">van Douwen [1980]</a> </p> |
| Conclusion | Statement |
|---|---|
| Form 146 | <p> \(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is an A1 space. (\((X,T)\) is A1 means if \(U \subseteq T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\) </p> |
The conclusion Form 135 \( \not \Rightarrow \) Form 146 then follows.
Finally, the
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 \(A\) |