Axiom Of Choice

Statement:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals has a choice function. Jech [1973b], p 148 prob 10.1.

Howard_Rubin_Number: 94

Parameter(s): This form does not depend on parameters

This form's transferability is: Transferable

This form's negation transferability is: Negation Transferable

Article Citations:
Sierpi'nski-1918: L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse

Book references
The Axiom of Choice, Jech, T., 1973b

Note connections:

The following forms are listed as conclusions of this form class in rfb1: 94, 5, 6, 13, 34, 35, 74, 194,

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Complete List of Equivalent Forms

Howard-Rubin Number	Statement	References
94 A	The discrete topology on \(\omega\) is Lindelöf.	Brunner [1982d] Note [40]
94 B	There is a Lindelöf space with a closed, discrete infinite subset. Brunner [1982d] and Note 40.	Brunner [1982d] Note [40]
94 C	\(\Bbb R\) is hereditarily Lindelöf.	Lindelof [1905] Note [40]
94 D	\({\Bbb R}\) is Lindelöf.	Young [1903] Herrlich-Strekcer-1997 Note [40]
94 E	Every second countable topological spaceis Lindelöf. G.Moore [1982] p 236.	Herrlich-Strekcer-1997 Note [40] Book: Zermelo's Axiom of Choice
94 F	For every \(A\subseteq{\Bbb R}\) and \(x\in{\Bbb R}\)the following definitions are equivalent: \(x\) is in the closure of \(A\) iff every neighborhood of \(x\)intersects \(A\). \(x\) is in the closure of \(A\) iff there is a sequence\(\{x_{n}\}\subseteq A\) such that \(\lim_{}x_{n}= x\). Jech [1973b] p 21. (See [94 S].)	Sierpiński [1918] Herrlich-Strekcer-1997 Note [40] Book: The Axiom of Choice
94 G	Every subset of \(\Bbb R\) is separable. Jech [1973b] p 142.	Herrlich-Strekcer-1997 Note [40] Book: The Axiom of Choice
94 H	Every subspace of a separable (pseudo)metric space isseparable. Jech [1973b] p 21 and Note 40.
94 I	Every separable metric space is Lindelöf.	Good-Tree-1995 Note [40]
94 J	Every second countable metric space is Lindelöf.	Good-Tree-1995 Note [40]
94 K	Every second countable (pseudo)metric space is separable. Keremedis [1998b].	Good-Tree-1995
94 L	\(\Bbb Q\) is Lindelöf.	Herrlich-Strekcer-1997
94 M	In \({\Bbb R}\) every unbounded set contains acountable unbounded set.	Herrlich-Strekcer-1997
94 N	Partial Choice for Countable Families of Sets ofReals: Every countable family of non-empty sets of real numbers hasan infinite subset with a choice function. Sierpinski [1916].and Fraenkel/Bar Hillel [1958].
94 O	For all \(A\subseteq{\Bbb R}\), \(x\in A\) and \(f: A\rightarrow{\Bbb R}\) the following are equivalent: \((\forall\epsilon>0)(\exists\delta>0)(\forall y\in A)(\|y - x\| <\delta\) implies \(\|f(y) - f(x)\|<\epsilon)\) Whenever \(\{x_{n}\}\subseteq A\)and \(\lim_{}x_{n} = x\) then \(\lim_{}f(x_{n}) = f(x)\). Note 5.	Herrlich-Strekcer-1997
94 P	\((\forall f: \Bbb R\rightarrow{\Bbb R})(\forall x\in{\Bbb R})\) the following are equivalent: \((\forall\epsilon > 0)(\exists\delta >0)((\forall y\in\Bbb R)(\|y - x\|<\delta\rightarrow \|f(y) - f(x)\|<\epsilon)\). Whenever \(\lim_{}x_{n} = x\), then\(\lim_{}f(x_{n})=f(x)\). (A real valued function on \(\Bbb R\) is continuous if and only if it is sequentially continuous.)	Sierpiński [1918] Herrlich-Strekcer-1997 Book: Naive Mengen und Abstracte Zahlen, Band III
94 Q	The Classical Ascoli Theorem. For any set \(F\) ofcontinuous functions from \(\Bbb R\) to \(\Bbb R\), the followingconditions are equivalent:\itemitem{(1)} Each sequence in \(F\) has a subsequence thatconverges continuously to some continuous function (notnecessarily in \(F\)).\itemitem{(2)} (a) For each \(x\in {\Bbb R}\), the set \(F(x) = \{f(x) : f\in F\}\) is bounded, and\itemitem{} (b) \(F\) is equicontinuous.\parRhineghost [2000] and Note 10 \item{}{\bf[94 R]} Weak Determinateness. If \(A\) is a subset of\({\Bbb N}^{\Bbb N}\) with the property that\newline\centerline{\((\forall a\in A)(\forall x\in {\Bbb N}^{\Bbb N})\left( x_n = a_n\hbox{ for } n=0 \hbox{ and } n \hbox{ odd } \tox\in A\right)\)}\newlineThen in the game \(G(A)\) one of the two players has a winningstrategy. Rhineghost [2000] and Note 153.
94 S	If \(X\subseteq\Bbb R\) and \(x\) is an accumulationpoint of \(X\), then \(x\) is a limit point of \(X\). Sierpi\'nski [1918], Felscher \cite{1979}, and Note 40. (See [94 F].)
94 T	There exists a non-compact Lindel\"of T\(_1\) space.H.~Herrlich [2001].
94 U	There exists a non-compact Lindel\"of subspace of\(\Bbb R\). H.~Herrlich [2001].
94 V	There exists an unbounded Lindel\"of subspace of\(\Bbb R\). H.~Herrlich [2001].
94 W	There exists a non-closed Lindel\"of subspace of\(\Bbb R\). H.~Herrlich [2001].
94 X	\(C(\aleph_0,\)dense subsets of \(\Bbb R\)): Everycountable family of dense subsets of \(\Bbb R\) has a choice function.Gutierres [2004].
94 Y	Every dense subset of \(\Bbb R\) is separable.Gutierres [2004].
94 Z	Every second countable topological space issuper second countable. Gutierres [2004] and Note 159.
94 AA	\(\Bbb R\) is super second countable.Gutierres [2004] and Note 159.
94 AB	Every separable metric space is supersecond countable. Gutierres [2004] and Note 159.
94 AC	Every separable pseudometric space is supersecond countable. \ac{Gutierres} and Note 159. \cite{2004}
94 AD	Every Linedl\"{o}f subspace of \({\mathbb R}\) issuper second countable. Gutierres [2004] and Note 159
94 AE	A second countable topological space is Hausdorff ifand only if every sequence has at most one limit. Gutierres [2004].

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