Statement:
\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
Howard_Rubin_Number: 126
Parameter(s): This form does not depend on parameters
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Brunner-1984d: Positive functionals and the axiom of choice
Blass-1979: Injectivity, projectivity and the axiom of choice
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 126 A | Every positive functional on a \(B\)-lattice iscontinuous. Brunner [1984d] and Note 16. |
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| 126 B | Form 131 +Form 350. Keremedis [1999a] andNote 132. |
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| 126 C | \(FSCB\) +Form 131.FSCB is``Every separable first countable space \((X,T)\) has adenumerable \(\pi\)-base.'' Keremedis [1999a] and Note 132. |
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| 126 D | For every denumerable family \(A\) of disjoint non-emptysets there is an infinite set \(C\subset\bigcup A\) such that for every\(a\in A\), \(0\le|C\cap a|<\omega\). Keremedis [1999a] and Note 132. |
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| 126 E | If \((X,T)\) is a topological space, then for everyfamily \(D=\{D_i: i\in\omega\}\) of dense open sets of\(X\), there is a countable dense set \(S\subseteq X\) such that for all\(i\in\omega\) and for all but finitely many \(s\in S\), \(s\in D_i\).Keremedis [1999a] and Note 132. |
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| 126 F | Every denumerable family of sets havingthe sfip also has a pseudo-intersection. Keremedis [1999a] andNote 132. |
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| 126 G | DUN(\(\omega\)): The countable disjoint union ofnormal topological spaces is normal.Howard/Keremedis/Rubin/Rubin [1998a] and Note 141. |
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| 126 H | DUU(\(\omega\)): If \(C\) is a countable collectionof pairwise disjoint topological spaces each of which satisfiesUrysohn's lemma, then the disjoint union of thespaces in \(C\) satisfies Urysohn's lemma.Howard/Keremedis/Rubin\slash Rubin [1998a] and Note 141. |
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| 126 I | DUT(\(\omega\)): If \(C\) is a countable collectionof pairwise disjoint topological spaces each of which satisfiesthe Tietze-Urysohn extension theorem, then the disjoint unionof the spaces in \(C\) satisfies the Tietze-Urysohn extension theorem.Howard/Keremedis/Rubin/Rubin [1998a] and Note 141. |
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| 126 J | \(F_\sigma\) subsets of paracompact spaces areparacompact. Howard/Kereme\-dis\slash Rubin/Rubin [1997b] andNote 141. |
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| 126 K | \(F_\sigma\) subsets of metacompact spaces aremetacompact. Howard/Kereme\-dis/Rubin/Rubin [1997b] andNote 141. |
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| 126 L | No maximal closed free filter in a \(T_1\)topological space has a countable filter base. Keremedis/Tachtsis [1999b] and Note 10. |
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| 126 M | Weierstrass compact pseudometric spaces arecountably compact. Keremedis [1999a] and notes 6 and10. |
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| 126 N | Weierstrass compact pseudometric spaces arecompact. Keremedis [1999a] and notes 6 and 10. |
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| 126 O | Every compact pseudometric space has a densesubset which can be written as a countable union of finite sets.Keremedis [1999a] and Note 10. |
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| 126 P | Every compact pseudometric space has a densesubset which can be written as a well ordered union of finitesets. Keremedis [1999a] and Note 10. |
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