Axiom Of Choice

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:

The following forms are listed as conclusions of this form class in rfb1: 94, 82, 185, 330, 350, 131,

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Complete List of Equivalent Forms

Howard-Rubin Number	Statement	References
126 A	Every positive functional on a $B$ -lattice iscontinuous. Brunner [1984d] and Note 16.
126 B	Form 131 +Form 350. Keremedis [1999a] andNote 132.
126 C	$FSCB$ +Form 131.FSCB is``Every separable first countable space $(X,T)$ has adenumerable $\pi$ -base.'' Keremedis [1999a] and Note 132.
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.
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.
126 F	Every denumerable family of sets havingthe sfip also has a pseudo-intersection. Keremedis [1999a] andNote 132.
126 G	DUN( $\omega$ ): The countable disjoint union ofnormal topological spaces is normal.Howard/Keremedis/Rubin/Rubin [1998a] and Note 141.
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.
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.
126 J	$F_\sigma$ subsets of paracompact spaces areparacompact. Howard/Kereme\-dis\slash Rubin/Rubin [1997b] andNote 141.
126 K	$F_\sigma$ subsets of metacompact spaces aremetacompact. Howard/Kereme\-dis/Rubin/Rubin [1997b] andNote 141.
126 L	No maximal closed free filter in a $T_1$ topological space has a countable filter base. Keremedis/Tachtsis [1999b] and Note 10.
126 M	Weierstrass compact pseudometric spaces arecountably compact. Keremedis [1999a] and notes 6 and10.
126 N	Weierstrass compact pseudometric spaces arecompact. Keremedis [1999a] and notes 6 and 10.
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.
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.

Form Classes

Complete List of Equivalent Forms