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\topmatter
\endtopmatter
Professor Kyriakos Keremedis has informed us of recent results showing that
forms 11, 12, 73, 75, 92, 337, 338, 339, and 360 are all equivalent to form
94. He has also pointed out some new equivalents of form 94.
This requires the following changes in the {\it Consequences of the
Axiom of Choice} Project:
\medskip
\noindent 1. {\bf Additions to Part VI, the Bibliography}
\smallskip
\noindent {\bf Herrlich, H. Strecker, E.}
\nopagebreak
\psc{\cite{1997}}{ {\it When is ${\Bbb N}$ Lindel\"of}, Comment\. Math\.
Univ\. Carolinae {\bf 38}, 553-556.}
\smallskip
\noindent{\bf Keremedis, K.}
\psc{\cite{1998}}{ {\it Countable disjoint unions and weak forms of AC},
preprint.}
\medskip
\noindent 2. {\bf Changes in Part II, the Topical List of Forms}
\smallskip
\noindent{$\bullet$} Make each of the changes below:
\smallskip
\noindent The old form 11 should be changed to
\item{}{\bf [94 C]} ${\Bbb R}$ is hereditarily Lindel\"of.
\ac{Lindel\"of} \cite{1905} and note 40.
\smallskip\noindent
form 12 should be changed to
\item{}{\bf [94 D]} ${\Bbb R}$ is Lindel\"of. \ac{Herrlich/Strecker}
\cite{1997}, \ac{Young} \cite{1903},and note 40.
\smallskip\noindent
form 360 should be changed to
\item{}{\bf [94 E]} Every second countable topological space
is Lindel\"of. \ac{Herrlich/Strecker} \cite{1997}, \ac{G.~Moore} \cite{1982}
p 236 and note 40.
\smallskip\noindent
form 73 should be changed to
\item{}{\bf [94 F]} For every $A\subseteq{\Bbb R}$ and $x\in{\Bbb R}$
the following definitions are equivalent:
\itemitem{(1)} $x$ is in the closure of $A$ iff every neighborhood of $x$
intersects $A$.
\itemitem{(2)} $x$ is in the closure of $A$ iff there is a sequence
$\{x_{n}\}\subseteq A$ such that $\lim_{}x_{n}= x$.
\item{} \ac{Herrlich/Strecker} \cite{1997}, \ac{Jech} \cite{1973b} p 21,
and note 40.
\smallskip\noindent
form 92 should be changed to
\item{}{\bf [94 G]} Every subset of ${\Bbb R}$ is separable.
\ac{Herrlich/Strecker} \cite{1997}, \ac{Jech} \cite{1973b} p 142,
and note 40.
\smallskip\noindent
form 75 should be changed to
\item{}{\bf [94 H]} Every subspace of a separable metric space is
separable. \ac{Jech} \cite{1973b} p 21 and note 40.
\smallskip\noindent
form 337 should be changed to
\item{}{\bf [94 I]} Every separable metric space is Lindel\"of.
\ac{Good/Tree} \cite{1995} and note 40.
\smallskip\noindent
form 338 should be changed to
\item{}{\bf [94 J]} Every second countable metric space is Lindel\"of.
\ac{Good/Tree} \cite{1995} and note 40.
\smallskip\noindent
form 339 should be changed to
\item{}{\bf [94 K]} Every second countable metric space is separable.
\ac{Good/Tree} \cite{1995} and \ac{Keremedis} \cite{1998}.
\smallskip\noindent
form [73 A] should be changed to
\item{}{\bf [94 O]} For all $A\subseteq{\Bbb R}$, $x\in A$ and $f: A
\rightarrow{\Bbb R}$ the following are equivalent:
\itemitem{(1)} $(\forall\epsilon>0)(\exists\delta>0)(\forall y\in A)
(|y - x| <\delta$ implies $|f(y) - f(x)|<\epsilon)$
\itemitem{(2)} Whenever $\{x_{n}\}\subseteq A$
and $\lim_{}x_{n} = x$ then $\lim_{}f(x_{n}) = f(x)$.
\item{} \ac{Herrlich/Strecker} \cite{1997} and note 5.
\smallskip\noindent
form [73 B] should be changed to
\item{}{\bf [94 P]}$(\forall f: {\Bbb R}\rightarrow{\Bbb R})(\forall
x\in{\Bbb R})$ the following are equivalent:
\itemitem{(1)} $(\forall\epsilon > 0)(\exists\delta >0)((\forall y\in{\Bbb R})
(|y - x|<\delta\rightarrow |f(y) - f(x)|<\epsilon)$
%\itemitem{(2)} Whenever $\lim_{}x_{n} = x$, then $\lim_{}f(x_{n})=f(x)$.
\item{} \ac{Herrlich/Strecker} \cite{1997}.
\medskip
\noindent{$\bullet$} Add the following new forms:
\smallskip
\item{}{\bf [94 L]} ${\Bbb Q}$ is Lindel\"of. \ac{Herrlich/Strecker}
\cite{1997}.
Put [94 L] in TOPOLOGICAL FORMS III. General Topology
\smallskip
\item{}{\bf [94 M]} In ${\Bbb R}$ every unbounded set contains a
countable unbounded set. \ac{Herrlich/Strecker} \cite{1997}.
Put [94 M] in TOPOLOGICAL FORMS VI. Properties of ${\Bbb R}$
\smallskip
\item{}{\bf [94 N]} Partial Choice for Countable Families of Sets of
Reals: Every countable family of non-empty sets of real numbers has
an infinite subset with a choice function. \ac{Sierpinski} \cite{1916}.
Put [94 N] in CHOICE FORMS IV. Partial Choice PC
\item{}{\bf [13 C]} Strong Bolzano-Weierstrass Theorem: In ${\Bbb R}$ for
every bounded, infinite set $A$, there is a convergent, injective sequence
whose terms are in $A$. \ac{Herrlich/Strecker} \cite{1997}.
\iput{Bolzano-Weierstrass theorem}
Put [13 A] in TOPOLOGICAL FORMS VI. Properties of ${\Bbb R}$\smallskip
\medskip
\noindent 3. {\bf Changes in Part I, Numerical List of Forms}
\smallskip
Delete forms 11, 12, 73, 75, 92, 337, 338, 339, and 360. Add the equivalents
of form 94 listed above and the equivalent of form 13.
\medskip
\noindent 4. {\bf Changes in Part III, Models}
\smallskip
Delete all references to forms 11, 12, 73, 92, 337, 338,
339, and 360. In the Cohen models section only 337, 338, and 339 occur and
they only occur in the description of M1. In the Fraenkel-Mostowski models
section 12 occurs only in the introduction. 11, 73, and 92 occur in the
introduction and in the descriptions of most of the FM models.
\medskip
\noindent 5. {\bf Changes in Part IV, Notes}
\smallskip
1. Eliminate notes 4, 6 and 10. (The results are in Herrlich/Strecker
[1997].)
2. Replace note 5 with and its table of contents entry with:
\noindent 5. A proof that [94 O] implies 74
\head{NOTE 5}\endhead [94 O] $\rightarrow$ form 74.
[94 O] is For all $A\subseteq{\Bbb R}$, $x\in A$ and $f: A
\rightarrow{\Bbb R}$ the following are equivalent:
\itemitem{(1)} $(\forall\epsilon>0)(\exists\delta>0)(\forall y\in A)
(|y - x| <\delta$ implies $|f(y) - f(x)|<\epsilon)$
\itemitem{(2)} Whenever $\{x_{n}\}\subseteq A$
and $\lim_{}x_{n} = x$ then $\lim_{}f(x_{n}) = f(x)$.\par
and form 74 is: For every
$A\subseteq\Bbb R$ the following are equivalent:
\itemitem{(1)} $A$ is closed and bounded.
\itemitem{(2)} Every sequence $\{x_{n}\}\subseteq A$ has a
convergent subsequence with limit in A.)\par
\demo{Proof} Assume [94 O]. The usual proof that part (1) of form 74
implies part (2) doesn't require any choice. So assume $A \subseteq
\Bbb R$ and $A$ satisfies (2) of 74 but not (1) of 74. Assume first that
$A$ is bounded. Then $A$ is not closed so there is an $x_{0} \not\in A$
such that every neighborhood of $x_{0}$ intersects $A$. Define $f : A
\cup \{x_{0}\} \rightarrow {\Bbb R}$ by $f(x) = 1$ if $x\in A$ and
$f(x)=0$ if $x = x_0$. Then (1) of [94 O] fails (with $\epsilon = 1$) so
there is a sequence $\{x_{n}\}\subseteq A\cup \{x_{0}\}$ such that $\lim_{}
f(x_{n})\neq f(x_{0})$. This means we can assume $\{x_{n}\}\subseteq A$
(replacing $\{x_{n}\}$ by a subsequence if necessary). By (2) of form 74,
$\lim_{} x_{n}\in A$ that is $x_0\in A$, a contradiction. Now assume that
$A$ is unbounded. Let $f$ be a monotone, increasing function from the
interval $(-1,1)$ one to one, onto ${\Bbb R}$ which satisfies
$(\forall a,b\in (-1,1))( | a - b| \le | f(a) - f(b)|$).
(For example, $f(x)={2x\over {1-x^{2}}}$ would work.) Then $f^{-1}(A)$
is bounded and satisfies (2) of 74 because if $\{x_{n}\}$ is a
sequence in $f^{-1}(A)$, then $\{f(x_{n})\}$ is a sequence in $A$.
Since $A$ satisfies (2) of 74, $\{f(x_{n})\}$ has a convergent
subsequence with limit in $A$. Using the property which we required
$f$ to have we infer that $f^{-1}$ applied to the convergent
subsequence mentioned above gives a convergent subsequence of
$\{x_{n}\}$ with limit in $f^{-1}(A)$. Since we have proved (2) of
74 implies (1) of 74 for bounded sets we conclude that
$f^{-1}(A)$ is closed. Hence there are numbers $a$ and $b$ such that
$-1 < a < b < 1$ and $(\forall x \in f^{-1}(A) )( a < x < b )$.
Since $f$ is monotone increasing $(\forall x \in A)( f(a) < x < f(b))$.
Hence $A$ is bounded. $\square$\enddemo
\par
3. Replace note 40 and its table of contents entry by the following:
\noindent 40. The equivalents of form 94
\head{NOTE 40}\endhead In this note we consider the equivalents
of form 94.
\ac{Brunner} \cite{1982d} proves that [94 B] implies form 13
(lemma 6, p. 164) by showing that the negation of form 13 implies
the negation of [94 A] which implies the negation of [94 B].
Since [94 A] clearly implies [94 B], it only remains
to prove that [94 A] implies 94 (to complete the argument for the
equivalence of 94, [94 A] and [94 B]). \ac{Herrlich/Strecker}
\cite{1997} have shown that forms 94 and [94 A] are equivalent.
In addition they show that forms [94 A],
[94 D], [94 E], [94 F], [94 G], [94 L], [94 M], [94 O] and [94 P]
are equivalent. It is clear that [94 E] implies [94 C] implies [94 D].
This answers a question of G\. \ac{Moore} \cite{1982} p 322:
Does [94 D] imply [94 C]? G\. Moore also asks if form 13 implies
[94 D]. This is also answered in the negative since 13 is
true and 94 false in $\Cal M6$.\par
It is shown in \ac{Keremedis} \cite{1998}
that [94 K] is equivalent to 94 and \ac{Sierpinski} \cite{1916} shows
the equivalence of [94 N] and [94]. \par
To get equivalence of [94 H], [94 I], and [94 J] to form 94,
we note first that (in ZF$^0$) every separable metric space is
second countable. From this it follows that [94 K] implies [94 H]
and [94 J] implies [94 I]. Combining these with the following
easy implications gives us the desired equivalences. [94 K] $\to$ [94 H],
[94 E] $\to$ [94 J], and [94 I] $\to$ [94 D]. Most of the above facts
were pointed out to us by K. Keremedis.
4. In note 18
\item{A}. Eliminate 11, 12, 73, and 92 from list 1: Forms that are
transferable.
\item{B}. Eliminate 11, 12, 73, and 92 from list 3: Forms that are true
in every permutation model. And in the parenthetical remark say "Each one
of these 49 forms is implied by one of the 12 forms 6, 37, 91, 130,
273, 305, 309, 313, 361, 363, 368, 369.)" (I've just eliminated 11 and 92
from the original list.)
5. In note 103 in the list following
" Similarly, the following forms are boundable
and therefore transferable:" eliminate forms 11, 12, 73 and 92
\medskip
\noindent 6. {\bf Changes in V, Reference for Table 1}
\smallskip
eliminate the entry at positions
$n$,$k$ where either $n$ or $k$ is 11, 12, 73, 75, 92, 337, 338, 339,
or 360. Add
\hskip.4in 94\ \ 74\ \ (1)\ \ note 5 \par
\end