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].

Form Classes

Complete List of Equivalent Forms